Mean Value THeorem

1.) The purple line has a slope that is defined as [f(b) - f(a)]/(b - a) which is parallel to the yellow line which has a slope defined as f '(c).

A,B and C are outputs or points of y = f(x). The line that A and B produces is the secant line in which, according to the Mean Value Theorem, is parallel to a tangent line that is somewhere between point A and B on the graph of f(x).

2.) The figure below displays a removable discontinuity and an corner.
There is a jump between x = 0 and x = 3. The blue line illustrates the slope between points A and B which clearly shows that the slope of A and B cannot have a parallel tangent on the function. There is a corner between x = 3 and x = 7. The purple line illustrates the slope between points C and D which is 0. x = 5 is the center between these two points and has a slope that is undefined. THEREFORE, the Mean Value Theorem only applies to continuous and differentiable functions.