# Let f in C^2 (i.e, f is differentiable twice and f′,f′′ are continuous. Show that f can be written as f(x)=g(x)+h(x) where g(x) is convex for any x and h(x) is concave for any x.

Let $f\in {\mathcal{C}}^{2}$ (i.e, $f$ is differentiable twice and ${f}^{\prime },{f}^{″}$ are continuous. Show that $f$ can be written as $f\left(x\right)=g\left(x\right)+h\left(x\right)$ where $g\left(x\right)$ is convex for any $x$ and $h\left(x\right)$ is concave for any $x$.
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Camden Stanton
Write ${f}^{″}$ as the sum of a (not necessarily continuous) non-negative function and non-positive function. Then integrate twice.