grandes-ecoles 2010 QIII.A.5

grandes-ecoles · France · centrale-maths1__mp Complex Numbers Argand & Loci Similarity, Rotation, and Geometric Transformations in the Complex Plane
The map $f\in\mathcal{E}$ satisfies $Tf = f$, $f([0,1]) = \tau$ where $\tau = \widehat{(-1)\,1\,\mathrm{i}}$, and $f(x) = -\overline{f(1-x)}$ for all $x\in[0,1]$.
a) Prove that $f$ is not injective (one may use the relation $f(1-x) = -\overline{f(x)}$).
b) More generally show that there exists no continuous bijection from $[0,1]$ onto $\tau$ (one may use an argument of arc-connectedness). 
The map $f\in\mathcal{E}$ satisfies $Tf = f$, $f([0,1]) = \tau$ where $\tau = \widehat{(-1)\,1\,\mathrm{i}}$, and $f(x) = -\overline{f(1-x)}$ for all $x\in[0,1]$.

a) Prove that $f$ is not injective (one may use the relation $f(1-x) = -\overline{f(x)}$).

b) More generally show that there exists no continuous bijection from $[0,1]$ onto $\tau$ (one may use an argument of arc-connectedness).
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