grandes-ecoles 2015 QIII.A.3

grandes-ecoles · France · centrale-maths2__psi Proof Proof That a Map Has a Specific Property

Let $f : C(0,1) \rightarrow \mathbb{R}$ be a continuous application. We define: $$u(x,y) = \begin{cases} \mathrm{N}_f(x,y) & \text{if } (x,y) \in D(0,1) \\ f(x,y) & \text{if } (x,y) \in C(0,1) \end{cases}$$ on $\bar{D}(0,1)$.
Prove that $u$ is an application continuous at every point of $C(0,1)$. What can be concluded about the application $u$?

Let $f : C(0,1) \rightarrow \mathbb{R}$ be a continuous application. We define:
$$u(x,y) = \begin{cases} \mathrm{N}_f(x,y) & \text{if } (x,y) \in D(0,1) \\ f(x,y) & \text{if } (x,y) \in C(0,1) \end{cases}$$
on $\bar{D}(0,1)$.

Prove that $u$ is an application continuous at every point of $C(0,1)$. What can be concluded about the application $u$?

Paper Questions

QI.A.1 QI.A.2 QI.B.1 QI.B.2 QI.B.3 QI.C.1 QI.C.2 QII.A QII.B.1 QII.B.2 QII.B.3 QII.C.1 QII.C.2 QII.D.1 QII.D.2 QII.D.3 QII.D.4 QIII.A.1 QIII.A.2 QIII.A.3 QIII.B.1 QIII.B.2 QIII.B.3 QIII.C QIV.A.1 QIV.A.2 QIV.A.3 QIV.A.4 QIV.B.1 QIV.B.2 QIV.B.3 QIV.C.1 QIV.C.2