$\mathbb R^n\supset[a,b]$ is domain of definition $f:[a,b]\to\mathbb R^n:t\mapsto(f^1(t),f^2(t),\ldots,f^n(t))$ where $f^i\in C^\infty$
I need to prove that image of $f$, that means $f[a,b]$, doesn't contain any open ball in it. Intuitively, if image contains an open ball, then $f$ has a lot of singular points in which it has no derivative. But I have difficulties with the proof of the statement. Can you help me?