continuity - Is it possible to have a function that has a first, but not a second, derivative at some point or throughout some interval?

Can a function be differentiable within some interval (or even at some point), but have no second derivative anywhere in that interval?

Intuitively, I'd have thought it isn't possible. If it doesn't have a second derivative anywhere in the interval, that tends to suggest the first differential is cuspy or discontinuous at all points.

That then implies

1. the original curve is differentiable at all points in the interval (we can exclude any sections of the interval where it isn't, as they're irrelevant and the Q can be answered while ignoring them), yet


2. while its derivative exists, it is cuspy/discontinuous at all points in the interval.

Is this possible?

• I'm ideally looking for an interval. But it occurs to me this might be an assumption. So I've opened it to "or a point" in case there is a way to achieve this at a single point but not within an interval.