grandes-ecoles 2014 QIVC

grandes-ecoles · France · centrale-maths1__mp Differential equations Higher-Order and Special DEs (Proof/Theory)

Let $M \in \mathbb{R}_+^* \cup \{+\infty\}$ and $f : {]-\infty, M[} \rightarrow \mathbb{R}$ be a continuous function such that $$\forall (x, y) \in {\left]-\infty, \frac{M}{2}\right[}^2, \quad 2f(x+y) = f(2x) + f(2y) \tag{IV.1}$$
Show that $f'' = 0$, then that the set of continuous solutions of equation (IV.1) forms an $\mathbb{R}$-vector space, for which we will determine a basis.

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

Show that $f'' = 0$, then that the set of continuous solutions of equation (IV.1) forms an $\mathbb{R}$-vector space, for which we will determine a basis.

Paper Questions

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