Let
$$f(x) = \begin{array}{cc}
x + a\sqrt{2}\sin x & , 0 \le x \lt \frac{\pi}{4} \\
2x\cot x + b & , \frac{\pi}{4} \le x \lt \frac{\pi}{2} \\
a\cos 2x - b\sin x & , \frac{\pi}{2} \le x \le \pi
\end{array}$$
If $f(x)$ is continuous for $0 \le x \le \pi$, then