Let \(f:U\to \mathbb{R}\) be real analytic, \(f(U)\subset V\) and \(g:V\to \mathbb{R}\) be real analytic. Then \(f\circ g\) is also real analytic. [1, Proposition 1.4.2]

The same is true for the multidimensional case [1, Proposition 2.2.8].

References Link to heading

1. S. Krantz and H. Parks, A Primer of Real Analytic Functions. Boston, MA: Birkhäuser Boston, 2002. doi:10.1007/978-0-8176-8134-0