Differentiable and Continuous Functions

We say that y=f(x)  is differentiable at a point x= x 0  if f ( x 0 )  is a finite number.

A function y=f(x)  is called differentiable if it is differentiable at every point where it is defined.

Lets consider a differentiable function y=f(x)  .Then

dy dx = lim Δx0 Δy Δx

f (x)= lim Δx0 Δy Δx

f (x) Δy Δx

The symbol ' ' is read as almost equal to.

Δy f (x)Δx

Now if Δx0  , then clearly Δy0  because f (x)  is a finite number. ( f(x)  is a differentiable function, so the derivative is finite) This would not be true if the derivative f (x)  was not a finite number.

What this says is that if y=f(x)  is a differentiable function, then little changes in x  will generate only little change in y . There will not be any large arbitrary shift in the value of y   when x  changes in value just slightly. That is precisely is the definition of a continuous function.

We can conclude that a differentiable function is a continuous function.

The converse is not true. Not all continuous functions are differentiable. You may consider the absolute value function y=f(x)=| x |  which is continuous everywhere, yet it is not differentiable everywhere. Actually, it is differentiable everywhere except at 0.