# a problem related to inverse function theorem

This problem is from Advanced Calculus 3rd by R. Buck.

Let $D$ be the unit disk, $x^2+y^2\leq1$. Consider a transformation $T$ of class $C^1$ on an open set containing $D$,

$$T:\begin{cases} u=f(x,y),\\ v=g(x,y), \end{cases}$$

whose Jacobian is never $0$ in $D$. Suppose that $T$ is near the identity map in the sense that $|T(p)-p|\leq\frac{1}{3}$ for all $p\in D$. Prove that there is a point $p_0$ with $T(p_0)=(0,0)^T$.

Thanks a lot.

Update: I have figured it out. Seems the problem does not involve too much "inverse function theorem". At first, I have no clue at all. Here is the idea.

Define a function $\varphi(p)=\|T(p)\|_2^2$. $\varphi(p)$ is in fact the distance function between $T(p)$ and $(0,0)^T$. Then we need to show $\min_{p\in D}\varphi(p)=0$. If this is true, then the claim is proved. Next, we show $p^*=\arg\min_{p\in D}\varphi(p)$ is an interior point of $D$. Then we can use the Fermat's condition and this gives $T(p^*)=0$ which completes the proof.

I deeply appreciate if someone can provide me examples of interesting applications of inverse/implicit function theorems.

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