Let $f$ be a piecewise regular real-valued function defined on some interval $[a,b]$, such that $f$ has only a finite number of discontinuities and extrema in $[a,b]$. Then the Fourier series of this function converges to $f$ when $f$ is continuous and to the arithmetic mean of the left-handed and right-handed limit of $f$ at a point where it is discontinuous.