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$\mathbb R^n\supset[a,b]$ 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?

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    Doesn't $f$ have the same image as the path $c:[0,1] \to \mathbb{R}^n, c(s)=f(a+s(b-a))$?2012-06-24

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