Question

Do points of inflection have to be differentiable?

Answer 1

That is a good question! I had to revisit the definition in the Calculus book by Stewart, which states:

My answer to your question is no, a function does not need to be differentiable at a point of inflection; for example, the piecewise defined function

`f(x)={(x^2,if x<0), (sqrt{x},if x ge0):}`

is concave upward on `(-infty,0)` and concave downward on `(0,infty)` and is continuous at `x=0`, so `(0,0)` is an inflection point but not differentiable there.