If $f(t)$ is periodic, is there any $t$ that would equal to DC components?

Question

Suppose $f(t)$ is periodic with period $T$. Would there be $t$ that would necessarily equal to DC component (it can be scaled)? By DC component, I mean $F(0)$ where $F$ is fourier coefficient of $f$.

Answer 1

Yes. Given a continuous function f(t) that is periodic with period T, and the Fourier transform of that function F(w), then there will necessarily always be some time t where F(0) == f(t). (That time t will be different for different functions f(t).) F(0) is also called the DC component, also called the average value of f(t). This can be proven using the intermediate value theorem, as AnonSubmitter85 pointed out.

The only way for a periodic function to not have such a point is for the periodic function to not be continuous. One simple example is a rectangle wave where

• g(t) = 1 for 0 <= t < T/3, and
• g(t) = 0 for T/3 <= t < T, and
• g(t) is periodic with period T.

The average value of such a square wave, G(0), is 1/3. But the function g(t) never exactly equals 1/3 -- it's value is always either 0 or 1 at all times, with discontinuities where it jumps from 0 to 1 and from 1 to 0.