Do points of inflection have to be differentiable?

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That is a amiable interrogation! I had to revisit the limitation in the Calculus compass by Stewart, which states:

My retort to your interrogation is no, a office does not insufficiency to be differentiable at a summit of bend; for sample, the piecewise defined office

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

is hollow upward on ##(-infty,0)## and hollow downward on ##(0,infty)## and is normal at ##x=0##, so ##(0,0)## is an bend summit but not differentiable there.

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