Are these two functions in $C^{\infty}$?

Question

Define the function $h(x)$ by setting $h(x) = e^{-1/x^2}$ if $x>0$ and $0$ if $x \leq 0$.

1. My question is how can I show $h(x)$ is in $C^{\infty}$?

2. Let $\| \mathbf{x} \| = \max_{i=1,2} | x_i |$. Then is $h(\| \cdot\|) : \mathbb{R}^2 \rightarrow \mathbb{R}$ in $C^{\infty}$ as well?

Answer 1

The only concern is at zero. All you need to do is show that all derivatives exist, and are zero, by considering the left and right derivatives.

For your question 2, remember that a two-variable function is differentiable at a point if both partial derivatives are continuous.

Answer 2

Hint for 2. Suppose $a>0.$ For $y>0,$ the function at $(a,a+y)$ equals $e^{-1/(a+y)^2}.$ But for $y<0$ the function is just the constant $e^{-1/a^2}.$ That doesn't look good for the existence of partial derivatives at $(a,a).$