The function f is defined by $\displaystyle f(x)=sinx\ for -\frac{\pi}{4}<x\leq\frac{\pi}{4}, and\ f(x+\frac{\pi}{2})=f(x)+\sqrt{2}$ $\displaystyle for\ all\ x \in \mathbb{R}$

$\displaystyle 1)\ Find\ an\ expression\ for\ f(x)\ for\ \frac{\pi}{4}<x\leq\frac{3\pi}{4}$

$\displaystyle 2)\ Prove\ that\ f\ is\ continuous\ at\ \frac{\pi}{4}.$ $\displaystyle That\ is,\ prove\ \lim_{x\rightarrow\frac{\pi}{4}}f(x)=f(\frac{\pi}{ 4})$