Do points of inflection have to be differentiable?

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.