TITLE: Is the function $f(x,y,z)=\sqrt{x^2 +y^2} +z^2$ smooth?
QUESTION [5 upvotes]: Let $f:\mathbb{R}^3 \rightarrow \mathbb{R}$ be defined by

$$f(x,y,z)=\sqrt{x^2 +y^2} +z^2 .$$

Is this function smooth? My head is telling me there should be a problem when $x=y=0$, but I'm not sure. Can anybody help me out?

REPLY [4 votes]: Your intuition is correct: Applying the definition gives that, e.g., the partial derivative $\frac{\partial f}{\partial x}$ does not even exist at $(0, 0, 0)$, and hence $f$ is not even differentiable once there, let alone smooth.