Question

check all the x - values at which
a. (1 point) f appears to be not continuous
□1 □2 □3 □4
b. (1 point) f appears to be not differentiable. for each x - value you choose, give a reason why f is not differentiable.
□1 reason:
□2 reason:
□3 reason:
□4 reason:

Explanation:

Step1: Recall continuity definition

A function is continuous at a point if the limit as $x$ approaches the point from the left equals the limit as $x$ approaches the point from the right and equals the function - value at that point.

Step2: Check $x = 1$

The function has a jump - discontinuity at $x = 1$. The left - hand limit and the right - hand limit are not equal.

Step3: Check $x = 2$

The function is smooth at $x = 2$, so it is continuous.

Step4: Check $x = 3$

The function is smooth at $x = 3$, so it is continuous.

Step5: Check $x = 4$

The function has a jump - discontinuity at $x = 4$. The left - hand limit and the right - hand limit are not equal.

Step6: Recall differentiability definition

A function is differentiable at a point if the left - hand derivative and the right - hand derivative exist and are equal.

Step7: Check non - differentiability at $x = 1$

Reason: There is a jump - discontinuity at $x = 1$. A function is not differentiable at a point of discontinuity.

Step8: Check non - differentiability at $x = 2$

Reason: There is a sharp corner at $x = 2$. The left - hand derivative and the right - hand derivative are not equal at a sharp corner.

Step9: Check non - differentiability at $x = 3$

The function is smooth at $x = 3$, so it is differentiable.

Step10: Check non - differentiability at $x = 4$

Reason: There is a jump - discontinuity at $x = 4$. A function is not differentiable at a point of discontinuity.

Answer:

a. $x = 1$, $x = 4$
b.
1 Reason: Jump - discontinuity
2 Reason: Sharp corner
4 Reason: Jump - discontinuity