Give an example of a function that is continuous at x = −2, x = 0 and x = 3 but whose derivative does not exist at any of those points

Question

The furthest I have gotten into this question is f(x) = |x+2| + |x| + |x-3| - 5

Answer 1

That's a good idea. The absolute value function has a cusp (corner) at which the function is continuous but not differentiable.

You probably need a piecewise linear function here, where f(x) = |x+2| for x <-1, then f(x) = |x| for -1 <= x < 1.75 and then f(x) = |x-3| for x>=1.75.

I found these domains by plotting the three functions separately on Desmos, then reading the domains from the graph.