Prove that the following function is $C^\infty$

Question

Prove that the following function is $C^\infty$ (and in the point $ξ=0$) :

$$f:\Bbb R\to\Bbb C:\xi\mapsto {e^{i\cdot\xi\cdot λ}-1\over i\cdot\xi}-λ$$ for whichever $$λ>0$$ I am trying to find a proof for this elementary problem that has been asked in a mathematical competition in France in 2004.Maybe can we find a proof?

Answer 1

An idea:

At $\;\xi=0\;$ :

$$\lim_{\xi\to0}f(\xi)\stackrel{l'Hospital}=\lim_{\xi\to0}\frac{i\lambda e^{i\lambda\xi}}i=\lambda$$

so that $\;\xi=0\;$ is a removable singularity. At any other point $\;f(\xi)\;$ is the quotient of two analytic functions and is thus analytic.