$f\left(x\right)$ is continuous in $\left[0, \pi \right]$

If $f\left(x\right)$ is continuous in $\left[a,b\right]$ that means it should be continuous at all points in $\left[a,b\right]$.

$\underset{x\to \frac{\pi }{4}}{\lim }f\left(x\right)=f\left(\frac{\pi }{4}\right)$

$\Rightarrow \frac{\pi }{4}+a=\frac{\pi }{2}+b$

$\Rightarrow a-b= \frac{\pi }{4}----\left(I\right)$

$\underset{x\to \frac{\pi }{2}}{\lim }f\left(x\right)=f\left(\frac{\pi }{2}\right)$

$\Rightarrow -a-b=b$

$\Rightarrow a=-2b---\left(II\right)$

From equation $\left(I\right) \& \left(II\right)$

$-3b = \frac{\pi }{4} \Rightarrow b = \frac{-\pi }{12}$

$\Rightarrow a = \frac{\pi }{6}$