Piecewise $C_1$ and piecewise continuous

Question

I would appreciate if the following questions could be clarified with your help.

1. If a function is piecewise $C_1$, does this imply that it's also piecewise continuous?

2. If a function is piecewise continuous, does this imply that it's also piecewise $C_1$?

3. Is the sawtooth function continuous? If so, it has sharp "bumps", why is it still considered continuous on its domain?

Answer 1

Any differentiable function is continuous. So 1 is clearly "yes". 2 is "no", there are lots of continuous functions which are not even differentiable at any point, the most famous examples of which are due to Weierstrass. For 3, I might recommend reviewing the definition of continuity; intuitively, a sawtooth wave can be drawn without picking one's pencil up off the paper. It is not differentiable at the corners, though.