Practice Questions for NET JRF Real Analysis Assignment: Continuity

1. Let \(f: X \rightarrow X\) be a function such that \(f(f(x)) = x\) for all \(x \in X\). Then

(a.) \(f\) is one-to-one and onto.
(b.) \(f\) is one-to-one but not onto.
(c.) \(f\) is onto but not one-to-one.
(d.) \(f\) need not be either one-to-one or onto.

2. A polynomial of odd degree with real coefficients must have

(a.) at least one real root.
(b.) no real root.
(c.) only real roots.
(d.) at least one root which is not real.

3. Consider the following sets of functions on \(\mathbb{R}\), \(W =\) The set of constant functions on \(\mathbb{R}\), \(X =\) The set of polynomial functions on \(\mathbb{R}\), \(Y =\) The set of continuous functions on \(\mathbb{R}\), \(Z =\) The set of all functions on \(\mathbb{R}\).

Which of these sets has the same cardinality as that of \(\mathbb{R}\)?

(a.) Only \(W\)
(b.) Only \(W\) and \(X\)
(c.) Only \(W\), \(X\), and \(Z\)
(d.) Only \(W\), \(X\), and \(Y\)

4. If \(f: [0,1] \rightarrow (0,1)\) is a continuous mapping, then which of the following is NOT true?

(a.) \(F \subseteq [0,1]\) is a closed set implies \(f(F)\) is closed in \(\mathbb{R}\).
(b.) If \(f(0) < f(1)\) then \(f([0,1])\) must be equal to \([f(0), f(1)]\).
(c.) There must exist \(x \in (0,1)\) such that \(f(x) = x\).
(d.) \(f([0,1]) \neq (0,1)\).

5. In which of the following cases, there is no continuous function \(f\) from the set \(S\) onto the set \(T\)?

(a.) \(S = [0,1]\), \(T = \mathbb{R}\)
(b.) \(S = (0,1)\), \(T = \mathbb{R}\)
(c.) \(S = (0,1)\), \(T = (0,1]\)
(d.) \(S = \mathbb{R}\), \(T = (0,1)\)

6. Let \(A_n \subseteq \mathbb{R}\) for \(n \geq 1\), and \(\chi_n: \mathbb{R} \rightarrow \{0,1\}\) be the function \(\chi_n(x) = \begin{cases} 0 & \text{if } x \notin A_n \\ 1 & \text{if } x \in A_n \end{cases}\), let \(g(x) = \lim_{n \rightarrow \infty} \sup \chi_n(x)\) and \(h(x) = \lim_{n \rightarrow \infty} \inf \chi_n(x)\).

Which of the following statements are true?

(a.) If \(g(x) = h(x) = 1\), then there exists \(m\) such that for all \(n \geq m\), we have \(x \in A_n\).
(b.) If \(g(x) = 1\) and \(h(x) = 0\), then there exists \(m\) such that for all \(n \geq m\), we have \(x \in A_n\).
(c.) If \(g(x) = 1\) and \(h(x) = 0\), then there exists a sequence \(n_1, n_2, \ldots\) of distinct integers such that \(x \in A_{n_k}\) for all \(k \geq 1\).
(d.) If \(g(x) = h(x) = 0\), then there exists \(m\) such that for all \(n \geq m\), we have \(x \notin A_n\).

7. Let \(f\) be a monotonically non-decreasing real-valued function on \(\mathbb{R}\). Then

(a.) \(\lim_{x \rightarrow a} f(x)\) exists at each point \(a\).
(b.) If \(a < b\), then \(\lim_{x \rightarrow a^{+}} f(x) \leq \lim_{x \rightarrow b^{-}} f(x)\).
(c.) \(f\) is an unbounded function.
(d.) The function \(g(x) = e^{-f(x)}\) is a bounded function.

8. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a strictly increasing continuous function. If \(\{a_n\}\) is a sequence in \([0,1]\), then the sequence \(\{f(a_n)\}\) is

(a.) Increasing.
(b.) Bounded.
(c.) Convergent.
(d.) Not necessarily bounded.

9. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined by \(f(x) = \left[x^2\right]\). The points of discontinuity of \(f\) are

(a.) Only the integral points.
(b.) All rational numbers.
(c.) \(\{\pm \sqrt{n} : n\) is a non-negative integer\(\}\).
(d.) All real numbers.

10. A monotonic function

(a.) Is always continuous.
(b.) Is continuous only if it has the intermediate value property.
(c.) Can be nowhere continuous.
(d.) Can be discontinuous at finitely many points.

11. Which one of the following functions has exactly two points of discontinuity?

(a.) \(f(x) = \begin{cases} 1 & \text{if } x > 0 \\ 0 & \text{if } x \leq 0 \end{cases}\)
(b.) \(f(x) = \begin{cases} 1 & \text{if } 0 \leq x \leq 1 \\ 0 & \text{otherwise} \end{cases}\)
(c.) \(f(x) = \begin{cases} x & \text{if } 0 \leq x \leq 1 \\ 0 & \text{otherwise} \end{cases}\)
(d.) \(f(x) = \begin{cases} 1 & \text{if } x \text{ is rational} \\ 0 & \text{if } x \text{ is irrational} \end{cases}\)

12. Let \(f: [a, b] \rightarrow \mathbb{R}\) be a continuous function and let \(f(a) < f(b)\). Then by the intermediate value theorem

(a.) \(f([a, b]) = [f(a), f(b)]\)
(b.) \(f([a, b]) \supseteq [f(a), f(b)]\)
(c.) \(f([a, b]) \subseteq [f(a), f(b)]\)
(d.) \(f([a, b]) \neq [f(a), f(b)]\)

13. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined by \(f(t) = t^2\) and let \(U\) be any non-empty open subset of \(\mathbb{R}\). Then

(a.) \(f(U)\) is open.
(b.) \(f^{-1}(U)\) is open.
(c.) \(f(U)\) is closed.
(d.) \(f^{-1}(U)\) is closed.

14. Let \(f:[0,1] \rightarrow \mathbb{R}\) be the continuous function defined by \(f(x) = \frac{(x-1)(x-2)}{(x-3)(x-4)}\). Then the maximal subset of \(\mathbb{R}\) on which \(f\) has a continuous extension is

(a.) \((-\infty, 3)\)
(b.) \((-\infty, 3) \cup (4, \infty)\)
(c.) \(\mathbb{R} \backslash \{3,4\}\)
(d.) \(\mathbb{R}\)

15. The continuous function \(f: \mathbb{R} \rightarrow \mathbb{R}\) defined by \(f(x) = (x^2 + 1)^{2003}\) is

(a.) Onto but not one-one.
(b.) One-one but not onto.
(c.) Both one-one and onto.
(d.) Neither one-one nor onto.

16. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a continuous function. If \(f(\mathbb{Q}) \subseteq \mathbb{N}\), then

(a.) \(f(\mathbb{R}) = \mathbb{N}\)
(b.) \(f(\mathbb{R}) \subseteq \mathbb{N}\) but \(f\) need not be constant.
(c.) \(f\) is unbounded.
(d.) \(f\) is a constant function.

17. Let \(f:[a, b] \rightarrow \mathbb{R}\) be a monotonic function. Then

(a.) \(f\) is continuous.
(b.) \(f\) is discontinuous at at most two points.
(c.) \(f\) is discontinuous at at finitely many points.
(d.) \(f\) is discontinuous at at most countable points.

18. Let \(f:[0,10) \rightarrow [0,10]\) be a continuous mapping. Then

(a.) \(f\) need not have any fixed point.
(b.) \(f\) has at least 10 fixed points.
(c.) \(f\) has at least 9 fixed points.
(d.) \(f\) has at least one fixed point.

19. Let \(S=[0,1] \cup [2,3)\) and let \(f: S \rightarrow \mathbb{R}\) be defined by \(f(x) = \left\{\begin{array}{ccc}2x & \text{ if } & x \in [0,1] \\ 8-2x & \text{ if } & x \in [2,3)\end{array}\right.\) If \(T = \{f(x): x \in S\}\), then the inverse function \(f^{-1}: T \rightarrow S\)

(a.) does NOT exist.
(b.) exists and is continuous.
(c.) exists and is NOT continuous.
(d.) exists and is monotonic.

20. Let \(f\) be a continuous function from \([0,4]\) to \([3,9]\). Then

(a.) There must be an \(x\) such that \(f(x) = 4\).
(b.) There must be an \(x\) such that \(3f(x) = 2x + 6\).
(c.) There must be an \(x\) such that \(2f(x) = 3x + 6\).
(d.) There must be an \(x\) such that \(f(x) = x\).

21. Let \(f_{1}\) and \(f_{2}\) be two real-valued functions defined on the real line. Define two functions \(g\) and \(h\) by \(g(x) = \max \{f_{1}(x), f_{2}(x)\}\) and \(h(x) = \min \{f_{1}(x), f_{2}(x)\}\). Then

\(g(x^{2}) + h(x)^{2} + 3g(x)h(x)\)

\(= f_{1}(x)^{2} + f_{2}(x)^{2} + 3f_{1}(x)f_{2}(x)\)

holds for all \(x \in \mathbb{R}\)

(a.) Always
(b.) Only if \(f_{1}(x) = f_{2}(x)\) for all \(x\) in \(\mathbb{R}\)
(c.) Only if \(f_{1}\) and \(f_{2}\) are both positive functions or both negative functions
(d.) Only if at least one of the functions \(f_{1}\) and \(f_{2}\) is identically zero.

22. Let \(f\) be a strictly monotonic continuous real-valued function defined on \([a, b]\) such that \(f(a) < a\) and \(f(b) > b\). Then which one of the following is TRUE?

(a.) There exists exactly one \(c \in (a, b)\) such that \(f(c) = c\)
(b.) There exist exactly two points \(c_{1}, c_{2} \in (a, b)\) such that \(f(c_{i}) = c_{i}\), \(i = 1,2\)
(c.) There exists no \(c \in (a, b)\) such that \(f(c) = c\)
(d.) There exist infinitely many points \(c \in (a, b)\) such that \(f(c) = c\)

23. Let \(X\) and \(Y\) be two non-empty sets and let \(f: X \rightarrow Y\) and \(g: Y \rightarrow X\) be two mappings. If both \(f\) and \(g\) are injective (i.e., one-to-one) then

(a.) \(X\) and \(Y\) must be infinite sets.
(b.) \(g = f^{-1}\) always.
(c.) One of \(f \circ g: Y \rightarrow X\) and \(g \circ f: X \rightarrow Y\) is always bijective (one-to-one and onto).
(d.) There exists a bijective mapping \(h: X \rightarrow Y\)

24. Define a function \(f\) on the real line by \(f(x)=\left\{\begin{array}{cc}x-[x]-\frac{1}{2} & \text { if } x \text { is not an integer, } \\ 0 & \text { if } x \text { is an integer }\end{array}\right.\) which of the following is true:

(a.) \(f\) is periodic with period 1, i.e., \(f(x+1) = f(x)\) for all \(x\)
(b.) \(f\) is continuous
(c.) \(f\) is one-to-one
(d.) \(\lim _{x \rightarrow a} f(x)\) exists for all \(a \in \mathbb{R}\)

25. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a function such that \(f(x+y) = f(x)f(y)\) for all \(x, y \in \mathbb{R}\) and \(f(x) = 1 + xg(x)\) where \(\lim _{x \rightarrow 0} g(x) = 1\). Then the function \(f(x)\) is

(a.) \(e^{x}\)
(b.) \(2^{x}\)
(c.) A non-constant polynomial
(d.) Equal to 1 for all \(x \in \mathbb{R}\)

26. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a continuous function satisfying \(f \circ f = f\). Then

(a.) \(f\) must be constant
(b.) \(f(x) = x\) for all \(x\) in the range of \(f\)
(c.) \(f\) must be a non-constant polynomial
(d.) There is no such function

27. Let \(f:[0,1] \rightarrow [0,1]\) be continuous and \(f(0)=0\), \(f(1)=1\). Then, \(f\) is necessarily

(a.) injective, but not surjective.
(b.) surjective, but not injective.
(c.) bijective.
(d.) surjective.

28. Let \(p\) be a polynomial of degree \(2n+1\) with real coefficients. We say that a real number \(a\) is a fixed point of \(p\) if \(p(a) = a\). Then, \(p\) has

(a.) exactly \(2n+1\) fixed points
(b.) at least one fixed point
(c.) at most one fixed point
(d.) \(n\) fixed points

29. Define the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) by \(f(x) = \left\{\begin{array}{cl}ax^{2}+b, & \text { if } x \leq 1 \\ cx+1, & \text { if } x > 1\end{array}\right.\) We want to find appropriate values of \(a, b\) and \(c\) such that

1. \(f\) is increasing in the interval \((0, \infty)\);

2. \(f'\) is continuous on \(\mathbb{R}\).

Which of the following is the correct statement about the values of \((a, b, c)\) for which both conditions (1) and (2) are satisfied?

(a.) \((3,2,6)\) is the only possible value
(b.) There are finitely many values of \((a, b, c)\)
(c.) \((-2,-3,-4)\) is one of the values
(d.) There are infinitely many values of \((a, b, c)\)

30. The limit \(\lim_{x \rightarrow 0} \frac{3^{x}-2^{x}}{4^{x}-3^{x}}\) equals

(a.) \(\frac{3}{4}\)
(b.) \(\log_{4/3}\left(\frac{3}{2}\right)\)
(c.) \(\log_{e}\left(\frac{3}{2}\right)\)
(d.) \(\frac{2}{3}\)

31. The equation \(x^2 = x\sin(x) + \cos(x)\) is true for

(a.) no real value of \(x\)
(b.) exactly one real value of \(x\)
(c.) exactly two real values of \(x\)
(d.) infinitely many real values of \(x\)

32. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be continuous with \(f(0) = f(1) = 0\). Which of the following is not possible?

(a.) \(f([0,1]) = \{0\}\)
(b.) \(f([0,1]) = [0,1)\)
(c.) \(f([0,1]) = [0,1]\)
(d.) \(f([0,1]) = \left[-\frac{1}{2}, \frac{1}{2}\right]\)

33. Consider the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) defined by \(f(x) = \left\{\begin{array}{cc}2x, & \text { if } x \text { is rational } \\ 3-x, & \text { if } x \text { is irrational }\end{array}\right.\) Then \(f\) is continuous

(a.) only at \(x = 1\)
(b.) only at \(x = 2\)
(c.) only at \(x = 0\)
(d.) at no point of \(\mathbb{R}\)

34. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined by \(f(x) = |x+1| - |x|\), then the range of \(f\) is

(a.) The whole of \(\mathbb{R}\)
(b.) The closed interval \([-1,1]\)
(c.) The unbounded subset of \(\mathbb{R}\)
(d.) None of the above

35. How many zeroes are there for the function \(x^2 - 5|x| + 6\)?

(a.) 3
(b.) 4
(c.) 6
(d.) None of these

36. Let \(f(x) = \cos^{-1} x\). Then \(f\) is one-one and onto if the domain and range are specified respectively as

(a.) \([-1,1]\) and \((-\infty, \infty)\)
(b.) \([-1,1]\) and \((0, 2\pi)\)
(c.) \([-1,1]\) and \((0, \frac{\pi}{2})\)
(d.) \([-1,1]\) and \([0, \pi]\)

37. A bijection is a map that is both one-one and onto. It is given that \(f: \mathbb{R} \rightarrow \mathbb{R}\) is not a bijection. Which of the following must be true?

(a.) If \(f\) is not one-one, then \(f\) is not onto.
(b.) If \(f\) is not onto, then \(f\) is not one-one.
(c.) \(f\) is neither one-one nor onto.
(d.) If \(f\) is one-one, then \(f\) is not onto.

38. Define \(f: \mathbb{R} \rightarrow \mathbb{R}\) as follows:

\(-f(x) = \left\{\begin{array}{cc}1, & \text { if } x \text { is rational } \\ \frac{\sin x}{x}, & \text { if } x \text { is irrational }\end{array}\right.\)

Then

(a.) \(f\) is continuous everywhere
(b.) \(f\) is continuous only at \(x = 0\).
(c.) \(f\) is continuous at all rational points.
(d.) \(f\) is continuous at all irrational points.

39. Let \(f, g: \mathbb{R} \rightarrow \mathbb{R}\) be continuous functions whose graphs do not intersect. Then for which function below the graph lies entirely on one side of the \(X\)-axis

(a.) \(f\)
(b.) \(g+f\)
(c.) \(g-f\)
(d.) \(g \cdot f\)

40. \(f(x) = e^x - e^{-x}\), \(g(x) = e^x + e^{-x}\). Then

(a.) Both \(f\) and \(g\) are even functions
(b.) Both \(f\) and \(g\) are odd functions
(c.) \(f\) is odd, \(g\) is even
(d.) \(f\) is even, \(g\) is odd

41. Let \(F: \mathbb{R} \rightarrow \mathbb{R}\) be a monotone function.

Then

(a.) \(F\) has no discontinuities.
(b.) \(F\) has only finitely many discontinuities.
(c.) \(F\) can have at most countably many discontinuities.
(d.) \(F\) can have uncountably many discontinuities.

42. Let \(X\) be a set and \(f, g: X \rightarrow X\) be functions. We can say that \(f \circ g\) is bijective if

(a.) at least one of \(f, g\) is bijective
(b.) both \(f\) and \(g\) are bijective
(c.) \(f\) is \(1-1\) and \(g\) is onto
(d.) \(f\) is onto and \(g\) is \(1-1\)

43. Let \(f, g: \mathbb{R} \rightarrow \mathbb{R}\) be functions. We can conclude that \(h(x) \leq f(x) \, \forall x \in \mathbb{R}\) if we define \(h: \mathbb{R} \rightarrow \mathbb{R}\) as

(a.) \(\min \{g(x), f(x)+g(x)\}\)
(b.) \(\min \{f(x), f(x)+g(x)\}\)
(c.) \(\max \{g(x), f(x)+g(x)\}\)
(d.) \(\max \{f(x), f(x)+g(x)\}\)

44. Let \(X\) be a non-empty set, \(f: X \rightarrow X\) be a function and let \(A, B \subset X\). Then the identity \(f(A \cap B) = f(A) \cap f(B)\) is

(a.) Always holds
(b.) holds if \(f\) is \(1-1\)
(c.) holds if \(f\) is onto
(d.) holds if \(A \cup B = X\)

45. The range of the function \(f(x) = \frac{x}{\sqrt{x^{2}+1}}, x \in \mathbb{R}\) is

(a.) \([-1,1]\)
(b.) \([-1,1)\)
(c.) \((-1,1]\)
(d.) None of these

46. Let \(f:(0, \infty) \rightarrow \mathbb{R}\) be uniformly continuous. Then

(a.) \(\lim _{x \rightarrow 0+} f(x)\) and \(\lim _{x \rightarrow \infty} f(x)\) exist
(b.) \(\lim _{x \rightarrow 0+} f(x)\) exists but \(\lim _{x \rightarrow \infty} f(x)\) need not exist
(c.) \(\lim _{x \rightarrow 0+} f(x)\) need not exist but \(\lim _{x \rightarrow \infty} f(x)\) exists
(d.) neither \(\lim _{x \rightarrow 0^{+}} f(x)\) nor \(\lim _{x \rightarrow \infty} f(x)\) need exist

47. Let \(S=\{f: \mathbb{R} \rightarrow \mathbb{R} \mid \exists \epsilon>0\) such that \(\forall \delta>0,|x-y|<\delta \Rightarrow|f(x)-f(y)|<\epsilon\}\). Then

(a.) \(S=\{f: \mathbb{R} \rightarrow \mathbb{R} \mid f \text{ is continuous }\}\)
(b.) \(S=\{f: \mathbb{R} \rightarrow \mathbb{R} \mid f \text{ is uniformly continuous }\}\)
(c.) \(S=\{f: \mathbb{R} \rightarrow \mathbb{R} \mid f \text{ is bounded }\}\)
(d.) \(S=\{f: \mathbb{R} \rightarrow \mathbb{R} \mid f \text{ is constant }\}\)

48. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a function satisfying \(f(x+y)=f(x) f(y), \forall x, y \in \mathbb{R}\) and \(\lim _{x \rightarrow 0} f(x)=1\). Which of the following are necessarily true?

(a.) \(f\) is strictly increasing
(b.) \(f\) is either constant or bounded
(c.) \(f(r x)=f(x)^{r}\) for every rational \(r \in \mathbb{Q}\)
(d.) \(f(x) \geq 0, \forall x \in \mathbb{R}\)