Affine function is a diffeomorphism?

Question

Given an affine function $f:\mathbb{R}^n \to \mathbb{R}^n$ defined for all $x\in \mathbb{R}^n$ by $$f(x)=T(x)+a$$ such that $T$ is an invertible Linear map and $a\in \mathbb{R}^n$, is $f$ a diffeomorphism?

Answer 1

It is a diffeomorphism iff $T$ is invertible.
It is easy to see that $f$ is invertible iff $T$ is invertible with inverse $$f^{-1}(x) = T^{-1}(x - a).$$

As $T^{-1}$ is a linear transformation, it is differentiable everywhere.