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$f(x) = \begin{cases} 1& \text{ if } -1\leq x < 0 \\ 1/2 & \text{ if } x = 0 \\ x& \text{ if } 0 < x \leq 1 \end{cases} \text{for the interval} [-1,1]$

So I understand that you suppose to calculate the integral for every region but here at x = 0 function is 1/2. So should I add the f(0) value? For example for calculating $a_0$

$a_0 =\int_{-1}^{0}1dx +\int_{0}^{1}xdx$

or should i add f(0) value

$a_0 =\int_{-1}^{0}1dx +\int_{0}^{1}xdx + f(0)$

borisbulut
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  • Please use MathJax https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference – Some Guy Feb 16 '21 at 22:20
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    At points of discontinuity, the Fourier series will converge to the mean of the left and right limits, so in this case, the series will converge to $1/2$ at $x=0.$ – mjw Feb 16 '21 at 22:22
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    The integral doesn't depend on the value of the integrand at one specific point. – herb steinberg Feb 16 '21 at 22:31
  • So am I suppose to ignore x= 0 point? Then how come series will converge to 1/2 at x = 0. This doesn't make sense to me. – borisbulut Feb 16 '21 at 22:47
  • Think about integral as area under graph of the function. In such setting "area" of the rectangle which has as a base "single" point and finite height is zero. – Salcio Feb 17 '21 at 02:05

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