Bump function, gluing diffeomorphisms

Question

Let $f(x)$ be the bump function $$ f(x_1,\dots,x_N)\triangleq \prod_{i=1}^N exp\left( \frac{1}{1-x_i^2} \right), $$ $I$ be the identity matrix and, $R$ be a special orthogonal matrix not equal to $-I$. Then is $$ x\mapsto (f(x)R + (1-f(x))I)\cdot x, $$ a $C^1(\mathbb{R}^N,\mathbb{R}^N)$-diffeomorphism (moreover, is it analytic)?