Question 1

Which of the following statements is always true about differentiability and continuity of a function $$f(x) at$$ $$x = a$$?

Accepted Answer

$$f(x) is $$differentiable at $$x = a$$ \implies f(x)$$ is continuous at x = a$

Question 2

A function $$f(x) is $$continuous for $$x = c$$ but not differentiable for some $$c$$ such that $$x = c. $$Which of the following could be true?

Accepted Answer

$$f(x)$$ has a sharp corner at $$x = c$$

Question 3

Let $$f(x) be $$defined as follows:

$$ f(x) = \begin{cases} x^2$, & x \u003c 1 \ 2x-1, & x \geq 1 \end{cases}

Is$$ $$f(x)$$ differentiable at $$x = 1$$?

Accepted Answer

No, because the left and right derivatives at $$x = 1$$ are not equal.

Question 4

Which of the following is a necessary condition for a function $$f(x) to be $$differentiable at $$x = a$$?

Accepted Answer

$$f(x) is $$continuous at $$x = a$$

Question 5

Suppose $$f(x) is $$continuous at $$x = 2$$ but not differentiable at $$x = 2. $$Which of the following is a possible explanation?

Accepted Answer

$$f(x)$$ has a cusp at $$x = 2$$

Question 6

If $$f(x) is $$differentiable at $$x = a$$, which of the following must be true?

Accepted Answer

$$f(x) is $$continuous at $$x = a$$

Question 7

Which of the following graphs represents a function that is continuous but not differentiable at $$x = 0$$?

Accepted Answer

A graph with a sharp corner at $$x = 0$$

Question 8

Let $$f(x) be a $$function such that $$\lim_{x 	o a^-} f(x) = \lim_{x 	o a^+} f(x) = f(a)$$, but $$\lim_{x 	o a^-} f'(x) 
eq \lim_{x 	o a^+} f'(x). $$What can be concluded about $$f(x) at$$ $$x = a$$?

Accepted Answer

$$f(x) is $$continuous but not differentiable at $$x = a$$

Question 9

Which of the following is NOT a reason for a function to fail to be differentiable at a point?

Accepted Answer

The function is continuous and smooth at the point.

Question 10

Suppose $$f(x) is $$defined as $$f(x) = |x|. At $$which value of $$x is$$ $$f(x)$$ not differentiable?

Accepted Answer

$$x = 0$$

Question 11

A function $$f(x) is $$defined as $$f(x) = x$$^{1/3}$$. Is f(x$$)$$ differentiable at x = 0$?

Accepted Answer

Yes, because the derivative exists and is finite at $$x = 0$$

Question 12

Which of the following statements is true?

Accepted Answer

Differentiability implies continuity, but continuity does not imply differentiability.

Question 13

If $$f(x) is $$not continuous at $$x = a$$, which of the following must be true?

Accepted Answer

$$f(x) is $$not differentiable at $$x = a$$

Question 14

A function $$f(x) is $$defined as $$f(x) = x^2$ for x$$ \u003c 1$$ and f(x) = 2x - 1$ for x$$ \geq 1$$. Is f(x$$)$$ continuous at x = 1$?

Accepted Answer

Yes, because $$\lim_{x 	o 1^-} f(x) = \lim_{x 	o 1^+} f(x) = f(1)$$

Question 15

Which of the following is an example of a function that is continuous everywhere but not differentiable at exactly one point?

Accepted Answer

$$f(x) = |x|$$

Question 16

Suppose $$f(x) is $$continuous at $$x = a$$ and $$\lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$$ does not exist. What can be concluded?

Accepted Answer

$$f(x) is $$not differentiable at $$x = a$$

Question 17

Which of the following is a real-world example where a function is continuous but not differentiable?

Accepted Answer

The velocity of a car as a function of time when the car suddenly changes direction.

Study Guide: Differentiability and Continuity in Calculus

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