grandes-ecoles 2010 QIII.B.2

grandes-ecoles · France · centrale-maths1__mp Complex Numbers Argand & Loci Similarity, Rotation, and Geometric Transformations in the Complex Plane

The integer part of the real number $x$ is denoted $[x]$. For every real $x$ and every non-zero natural number $n$: $r_n(x) = [2^n x] - 2[2^{n-1}x]$. The map $f\in\mathcal{E}$ satisfies $Tf = f$ and $f([0,1]) = \tau$.
Let $x\in[0,1]$.
a) If $x\in[0,1[$, by choosing: $$\alpha_n = \frac{r_1(x)}{2} + \cdots + \frac{r_n(x)}{2^n} \text{ and } \beta_n = \alpha_n + \sum_{k=n+1}^{\infty}\frac{1}{2^k},$$ prove that $f$ is not differentiable at $x$.
b) Prove that $f$ is not differentiable at $1$.

The integer part of the real number $x$ is denoted $[x]$. For every real $x$ and every non-zero natural number $n$: $r_n(x) = [2^n x] - 2[2^{n-1}x]$. The map $f\in\mathcal{E}$ satisfies $Tf = f$ and $f([0,1]) = \tau$.

Let $x\in[0,1]$.

a) If $x\in[0,1[$, by choosing:
$$\alpha_n = \frac{r_1(x)}{2} + \cdots + \frac{r_n(x)}{2^n} \text{ and } \beta_n = \alpha_n + \sum_{k=n+1}^{\infty}\frac{1}{2^k},$$
prove that $f$ is not differentiable at $x$.

b) Prove that $f$ is not differentiable at $1$.

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.B.1 QI.B.2 QI.B.3 QII.1 QII.2 QII.3 QII.4 QIII.A.1 QIII.A.2 QIII.A.3 QIII.A.4 QIII.A.5 QIII.A.6 QIII.B.1 QIII.B.2