a) Let $D$ be a domain whose boundary $C$ contains a straight-line segment $L$. Let $f(z)$ be analytic in $D$ and continous on $L$. Assume also that $\Im(f) = v(x,y)$ vanishes on $L$. Prove that $f$ is analytic on $L$.
I am a bit confused with what it means to be analytic on a line. I know how to show analyticity in a domain, via Morera, but for a line, I am not so sure.
b) Show that there is no function $f(z)$ analytic for $y>0$ and continuous for $y\geq0$ such that $$ f(z) = |x| $$