Fourier series of $f(x)$ and its convergence

Question

sorry for the inappropriate format. I'll edit asap. question is about convergence of series. Can you explain why is "f(x)=1 ,f(x)=1/2" please.

Answer 1

Since $f$ is piecewise smooth, its Fourier series converges at $x=a$ to the average of the left and right hand limits of $f$ at $x=a$. If $f$ is continuous at $x=a$, then this says the Fourier series of $f$ converges pointwise to $f(a)$.

Since $f(x)={1\over 2}$ on $-\pi<x<0$ and this is continuous, the Fourier series also converges pointwise to ${1\over 2}$ on $-\pi<x<0$.

Similarly, the Fourier series converges pointwise to $1$ on $0<x<\pi$.

More interesting is that we can also deduce the value of the Fourier series at $x=\pm\pi,0$ from the result above.