Suppose all Dirichlet conditions are met and we have a function that has jump discontinuities. Dirichlet's theorem says that the series converges to the midpoint of the values at the jump discontinuity. What bothers me then is: Dirichlet's theorem is basically telling us the series isn't the same as the function, precisely because it converges to the midpoint and doesn't itself have that jump discontinuity! So, are we being easy with the equality sign? Whats going on? Is a function really equal to its Fourier series representation?