Question

1. a) where is f(x) continuous but not differentiable? b) where else is f(x) not differentiable?

Explanation:

Step1: Recall continuity and differentiability criteria

A function is continuous if there are no breaks, jumps or holes. It is not differentiable at a point if there is a sharp - corner, vertical tangent or discontinuity.

Step2: Identify sharp - corner for part a

At $x=-2$, the function $y = f(x)$ has a sharp - corner. The function is continuous at $x = - 2$ (no break in the graph), but the slope changes abruptly, so it is not differentiable.

Step3: Identify other non - differentiable points for part b

At $x = 2$, there is a vertical asymptote, so the function is not continuous and thus not differentiable. Also, at the open - hole points (e.g., if there are any points where the function is not defined in the graph), the function is not continuous and not differentiable. In this graph, at $x=3$ and $x = 5$ there are holes, so the function is not continuous and not differentiable.

Answer:

a) $x=-2$
b) $x = 2,x=3,x = 5$