# The definition of continuously differentiable functions

When we say $f \in C^1$, we mean that f is continuously differentiable. Isn't the continuity a redundant word? I mean, we have a theorem that says if $f$ is differentiable then it is continuous. So why in most of the textbooks they always mention them two?

So these are all equivalent:

• $f \in C^1$
• $f$ is continuously differentiable
• $f'$ exists
• A function $f$ is continuously differentiable if $f$ is differentiable and its derivative is continuous. No, those are not all equivalent. The first two are; the third is weaker. The first two require that $f'$ not merely exist, but also be continuous. – Brian M. Scott Jan 24 '15 at 5:57
• Note the ending "-ly", which makes it an adverb, not an adjective. So "continuously differentiable" means "differentiable in a continuous way". – Hans Lundmark Jan 24 '15 at 9:22

A function is said to be differentiable at a point if the limit which defines the derivate exists at that point. However, the function you get as an expression for the derivative itself may not be continuous at that point. A good example of such a function is $$x^2(\sin(1/x^2),$$ which has a finite derivative at $$x=0,$$ but the derivative is essentially discontinuous at $$x=0.$$
A continuously differentiable function $$f(x)$$ is a function whose derivative function $$f'(x)$$ is also continuous at the point in question.