$f : \mathbb{R}^n\rightarrow\mathbb{R}^n$ is smooth. Prove: if its tangent maps are orthogonal transformations, then f is orthogonal transformation...

Question

This is a problem that I meet in manifold class. At the beginning, I want to show $\left <f(x),f(x)\right >$, $\forall x \in\mathbb{R}^n$, and then use Taylor's theorem with remainder, but I found that it is possibly incorrect since I meet the following problem:

(1) Taylor's theorem has intergral terms. Since when make multiplication, I meet problem likes that $\int^{1}_{0}f(x)g(x)dx\neq \int_{0}^{1}f(x)dx\int_{0}^{1}g(x)dx$.

(2) The extension has constant term that can't elimate. Out of this, I can't think of any other way. Please help with my question, thanks for you reading.