grandes-ecoles 2014 QIVB

grandes-ecoles · France · centrale-maths1__mp Differentiating Transcendental Functions Regularity and smoothness of transcendental functions

Let $M \in \mathbb{R}_+^* \cup \{+\infty\}$ and $f : {]-\infty, M[} \rightarrow \mathbb{R}$ be a continuous function such that $$\forall (x, y) \in {\left]-\infty, \frac{M}{2}\right[}^2, \quad 2f(x+y) = f(2x) + f(2y) \tag{IV.1}$$
Deduce that the function $f$ is of class $C^\infty$ on $]-\infty, M[$.

Let $M \in \mathbb{R}_+^* \cup \{+\infty\}$ and $f : {]-\infty, M[} \rightarrow \mathbb{R}$ be a continuous function such that
$$\forall (x, y) \in {\left]-\infty, \frac{M}{2}\right[}^2, \quad 2f(x+y) = f(2x) + f(2y) \tag{IV.1}$$

Deduce that the function $f$ is of class $C^\infty$ on $]-\infty, M[$.

Paper Questions

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