Consider this function, defined on the number line for all real numbers / 0 if x is irrational, | f(x) = < | \ 1 if x is rational. This function is discontinued at every point on the number line. Next consider function g(x) = 1 - f(x). This function is discontinued at every point on the number line, too. But the sum of these two functions f(x) +g(x) = f(x) + (1 - f(x)) == 1 is identically equal to 1 in all number line and is, therefore, continue function in all points of the number line.