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Let $f: S^n\to S^n$ be a continuous map such that $f(p)=q$ and there exists neighborhoods of $p$ and $q$ such that $f$ is linear and invertible between these two neighborhoods, I wonder if $f$ is also globally linear and invertible on $S^n$ since intuitively it looks true.

By "locally linear", I mean if we identify $S^n$ with the simplex $\Delta^n$, then the induce map betweem two $\Delta^n$ is locally linear. This notion comes from Hatcher's algebraic topology, page 359, Ex.15, let me know if I interpret it wrong.

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    I am still unsure what locally linear is supposed to mean ... – 2017-02-16
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    At any rate, if $f\colon \Bbb R^{n+1}\to\Bbb R^{n+1}$ is a linear map and for some open set $U$ with $U\cap S^n\ne\emptyset$ we have $f(U\cap S^n)\subset S^n$, then $f$ must be orthogonal. – 2017-02-16
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    @HagenvonEitzen Thank you! But how to prove this fact you mentioned? – 2017-02-16
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    The word combination "locally linear" does not occur in the text you link. And Ex 14 on page 359 reads: "Use cellular approximation to show that the $n$ skeletons of homotopy equivalent CW complexes without cells of dimension $n + 1$ are also homotopy equivalent." – 2017-02-16
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    @HagenvonEitzen I am so sorry, it should be Ex.15 – 2017-02-16
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    @HagenvonEitzen I actually misunderstood that problem and the statement in my question may not be true I think. – 2017-02-16

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