Multiple choice (Continuity and differentiability)

Choose and write the correct option in the following question:(continuity and differentiability multiple choice)

1.) The function f:R → R given by f(x) = -|x – 1| is

(a) Continuous as well as differentiable at x = 1

(b) Not continuous but differentiable at x = 1

(d) Neither continuous nor differentiable at x = 1

Answer (c)

2.) The function $f(x) = e^{|x|}$ is

(a) Continuous everywhere but not differentiable at x = 1

(b) Continuous and differentiable everywhere

(d) None of these

Answer (a)

3.) The function f(x) = [x], where [x] denotes the greatest integer function, is continuous at

(a) 4 (b) 2

Answer (d)

4.) The number of points at which the function $f(x) = \frac{1}{x-[x]}$ is not continuous is

(a) 1 (b) 2

Answer (d)

5.) The function $f(x) = \left\{\begin{array}{l} -\frac{\sin x}{x}+\cos x, \text { if } x \neq 0 \\ k, \text { if } x = 0 \end{array}\right\}$ is continuous at x = 0 then the value of k is

(a) 3 (b) 2

Answer (b)

6.) If the function f defined by $f(x) = \left\{\begin{array}{l} -\frac{k\cos x}{\pi - 2x}, \text { if } x \neq \frac{\pi}{2} \\ 3, \text { if } x = \frac{\pi}{2} \end{array}\right\}$ , is continuous at x =π/2, then the value of

(a) 2 (b) 3

Answer (c)

7.) The function f(x) = cot x is discontinuous on the set

(a) {x = nπ: n∈ Z} (b) {x = 2nπ: n∈ Z}

Answer (a)

8.)Let f(x) = |sin x|. then

(a) f is everywhere differentiable.

(b) f is everywhere continuous but not differentiable at x = nπ: n∈Z

(d) None of these

Answer (b)

9.) The function $f(x) = \frac{x-1}{x(x^2-1)}$ is discontinuous at

(a) Exactly one point (b) Exactly two points

Answer (c)

10.) If $f(x) = x^2\sin\frac{1}{x}$ , where x ≠ 0, thenthe value of the function f at x = 0, so that the function is

(a) 0 (b) -1

Answer (a)

11.) The function $f(x) =\left\{\begin{array}{l} \frac{e^{3x}-e^{-5x}}{x}, \text { if } x \neq 0 \\ 3, \text { if } x = 0 \end{array}\right\}$ is continuous at x = 0 for the value of k, as