Given $f(x+T) = f(x) + a$, prove that $f(x) = \varphi(x) + {a \over T}x$, where $\varphi(x)$ is periodic with period $T$
Let $\phi(x)=f(x)-{a\over T}x$, $\phi(x+T)=f(x+T)-{a\over T}(x+T)=f(x)+a-{a\over T}(x+T)=f(x)-{a\over T}x=\phi(x)$.
Given $f(x+T) = f(x) + a$, prove that $f(x) = \varphi(x) + {a \over T}x$, where $\varphi(x)$ is periodic with period $T$
Let $\phi(x)=f(x)-{a\over T}x$, $\phi(x+T)=f(x+T)-{a\over T}(x+T)=f(x)+a-{a\over T}(x+T)=f(x)-{a\over T}x=\phi(x)$.