Let $(V,\lvert\cdot\rvert)$ be a normed vector space, $U$ an open subset of $V$, and $f:U\rightarrow V$. Suppose $f$ is differentiable at $\xi$ and $Df(\xi)$ is invertible. Show there exists a neighborhood $N$ of $\xi$ on which $f$ is injective.

Proof$\quad$ Put $T=Df(\xi)$. Since $T$ is invertible, $\lVert T^{-1}\rVert\not=0$, and $$\lvert x-y\rvert=\lvert T^{-1}(Tx-Ty)\rvert\leq\lVert T^{-1}\rVert\lvert Tx-Ty\rvert\mbox{,}$$ so that $$\frac{\lvert x-y\rvert}{\lVert T^{-1}\rVert}\leq\lvert Tx-Ty\rvert $$ for any $x,y\in V$.

$\qquad$ Since $f$ is differentiable at $\xi$, we have $$f(x)=f(\xi)+T(x-\xi)+\lvert x-\xi\rvert R(x-\xi)\mbox{,} $$ where $R(x-\xi)\rightarrow 0$ as $x\rightarrow \xi$. Choose $\delta>0$ so that $\vert x-\xi\rvert<\delta$ implies $\lvert R(x-\xi)\rvert<\frac{1}{2\lVert T^{-1}\rVert}$. It follows then that for $\lvert x-\xi\rvert<\delta$, $\lvert y-\xi\rvert<\delta$, $$ \begin{aligned} \lvert f(x)-f(y)\rvert &= \lvert Tx-Ty+\lvert x-\xi\rvert R(x-\xi)+\lvert y-\xi\rvert R(y-\xi)\rvert\\ &\geq \lvert Tx-Ty\rvert-\lvert x-\xi\rvert\lvert R(x-\xi)\rvert-\lvert y-\xi\rvert\lvert R(y-\xi)\rvert \\ &> \frac{\vert x-y\rvert}{\lVert T^{-1}\rVert}-(\lvert x-\xi\rvert+\lvert y-\xi\rvert)\frac{1}{2\lVert T^{-1}\rVert}\\ &> \frac{\lvert x-y\rvert}{2\lVert T^{-1}\rVert}\mbox{.} \end{aligned} $$

Is this correct?

  • $\begingroup$ You have have shown that $f$ is injective on $B(\xi,\delta)$, the ball with radius $\delta$ centered in $\xi$, right? $\endgroup$
    – Filippo
    Nov 24 '21 at 7:10
  • 1
    $\begingroup$ How did you go from the second last line to the last line in the equation above? I think you have a mistake in your estimate, we have $|x -\xi|+|y-\xi| \ge |x-y|$, not the other way around. $\endgroup$
    – copper.hat
    Nov 25 '21 at 4:33

This is not true.

Let $f(x) = {x \over 2} + x^2 \sin {1 \over x}$ for $x \neq 0$ and $f(0) = 0$. Note that $f$ is smooth for $x \neq 0$. We see that $f'(0) = {1 \over 2} >0$ and for $x \neq 0$ we have $f'(x) = {1 \over 2} + 2x \sin {1 \over x} - \cos {1 \over x}$.

We see that there are points $x_+,x_- \neq 0$ arbitrarily close to $0$ such that $f'(x_+)>0$ and $f'(x_-)<0$. Since $f'$ is continuous, this means that $f$ is increasing locally around $x_+$ and decreasing locally around $x_-$. In particular, $f$ cannot be locally injective at $x=0$.

This is slightly counterintuitive, maybe because we imagine derivatives to be continuous.

A related result is found in Tao's answer which is an answer to his question on MO here. The difference is (apart from $V=\mathbb{R}^n$) that $f$ is assumed to have an invertible derivative everywhere. (And frankly, that this results in a positive answer is very surprising to me and highlights my lack of understanding things global. Tao's answer is fairly involved.)


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