Show that the curve $x^2-y^2=0$ is not smooth in any neighborhood of the point(0,0). The formal definition of a smooth curve is: "A set $S$ is a smooth curve if
(a) $S$ is connected, and
(b) every $a\in S$ has a neighborhood $\mathcal{N}$ such $S$ and $\mathcal{N}$ is the graph of a $C^1$ function $f$."
The question seems obvious if we just look at the graph, but how do we explain it through the definition of smooth curve?
First, this curve is union of the lines $y=x$ and $y=-x$. Let's pick $a=(0,0)$. For any neighborhood $N$ around $a$, we can apply the vertical line test to see that $x^2-y^2=0$ is not even a function so this obviously violates (b).