A question about inflection point Does the function
$$
f(x)=
\begin{cases}
x^2   & x\leq1\\
2-x^2 & x>1
\end{cases}
$$
has an inflection point at $x=1$, or it hasn't because $f$ isn't differentiable at $x=1$? Thanks! 

Surprise surprise! A function doesn't have to be differentiable in order to have an inflection point. Simply put, an inflection point is a point at which $f(x)$ switches concavity. 
$\displaystyle \lim_{x \to 1^-}f''(x)=2$ and $\displaystyle \lim_{x \to 1^+}f''(x)=-2$
We can easily see that $f(x)$ is continuous at $x=1$.
Because the concavity changes signs at $x=1$, yes, $f(x)$ has a point of inflection.
