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 criteria

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

Step2: Analyze the graph for discontinuity

At $x = 1$, the function has a jump - the left - hand limit and the right - hand limit are not equal. At $x = 3$, the function has a hole, so the limit as $x$ approaches 3 does not equal the function value (since the function is not defined at the exact point of the hole).

Step3: Recall differentiability criteria

A function is differentiable at a point if it is continuous at that point and the left - hand derivative equals the right - hand derivative.

Step4: Analyze the graph for non - differentiability

At $x = 2$, there is a sharp turn (cusp), so the left - hand derivative and the right - hand derivative are not equal. At $x = 4$, the function is continuous but has a vertical tangent (the slope of the tangent line becomes infinite), so it is not differentiable.

Answer:

a. 1, 3
b. 2 Reason: Sharp turn (cusp), so left - hand and right - hand derivatives are not equal.
4 Reason: Vertical tangent, so the derivative is infinite.