Consider a function f(x) = cos(1/x) if x not 0 and f(0) = 0. Then it is continuous everywhere except the point x = 0. For all x not 0 the limit of f(x+h) -f(x-h) = 0 (as h to 0) because of the continuity of f. We also have that f(0+h) - f(0-h) = f(h) - f(-h) = 0 and the limit for x = 0 is zero, too. However, the function is not continuous at x = 0.