a) Show that the set of functions S= {$\frac {sin(nx)}n$}$_{n\ge1}$ on $I=[0,1]$ is equicontinuous and uniformly bounded.
b) Show that the IVP $y'=f(x,y)$, $y(1)=1$
with $f(x,y)$ = $\begin{cases} x, x\le1 \\[2ex] 2x, x\gt1 \end{cases}$
does not have a solution on any interval of the form $[1-\varepsilon, 1+ \varepsilon]$ for all $\varepsilon$ satisfying $\varepsilon\gt0$ ie there is no differentiable funciton on this interval satisfying the IVP.
Here is (b). Assume $g$ is a differentiable function solving this IVP. Choose $h\in (0,\epsilon)$. By MVT find $c\in(1,1+h)$ such that $$\frac{g(1+h)-g(1)}{h}=g'(c)=f(c,g(c))=2c$$ Similarly, establish $d\in (1-h,1)$ so that $$\frac{g(1)-g(1-h)}{h}=g'(d)=f(d,g(d))=d$$ Therefore $$\Bigg|\frac{g(1+h)-g(1)}{h}-\frac{g(1)-h(1-h)}{h}\Bigg|=2c-d\geq2(1)-1=1$$ which is a contradiction.