A continuous map whose graph is an embedded submanifold is smooth?

Let $M$ and $N$ be smooth manifolds and $f: M \to N$ a continuous map. Suppose the graph of $f$ is an embedded submanifold of $M \times N$. Is it true that $f$ is smooth?

Consider the map $x \mapsto x^{\frac{1}{3}}$ from the real line to itself.