The function $f ( x ) = \left\{ \begin{array} { l l } 0 & \text { if } x 0 \end{array} \right\}$ has a Fourier series on $[ - 2,2 ]$ that converges at $x = 0$ to A) 0 B) 1 C) 1/2 D) $- 1$ E) unknown

Question

The function $f ( x ) = \left\{ \begin{array} { l l } 
0 & \text { if } x < 0 \
1 & \text { if } x > 0
\end{array} \right\}$ has a Fourier series on $[ - 2,2 ]$ that converges at $x = 0$ to
A) 0
B) 1
C) 1/2
D) $- 1$
E) unknown

Quizplus · Accepted Answer

The Fourier series of a function converges to the average of the left-hand and right-hand limits at points of discontinuity. At $x = 0$, the left-hand limit of $f(x)$ is 0 and the right-hand limit is 1. Therefore, the Fourier series converges to $(0 + 1)/2 = 1/2$ at $x = 0$.

The Function f(x)={0 if x<01 if x>0}f ( x ) = \left\{ \begin{array} { l l } 0 & \text { if } x < 0 \\1 & \text { if } x > 0\end{array} \right\}f(x)={01​ if x<0 if x>0​}

The Function $f ( x ) = \left\{ \begin{array} { l l } 0 & \text { if } x < 0 \\1 & \text { if } x > 0\end{array} \right\}$