Differentiation III: How Differentiability and Continuity Are Related

Primary tabs

Lesson Summary: 
We discuss here how a function is continuous at a point if it is differentiable at that point. However, the converse is not true. If a function is continuous at a point it is not guaranteed that it will be differentiable there.
Lesson Inputs: 
Function Limits VI: Continuous Functions
Differentiation II: The Gory Details of Calculating Derivatives
Lesson Outputs: 
Differentiation IV: Differentiation Formulas Everyone Must Know

SAMPLE LESSON

Sign in for easy to read lessons narrowly focused on a specific subject. All lessons are linked to each other allowing you to easily see the connections between different Calculus concepts. Our unique Calcmap will give you a bird's-eye view of all our lessons and help you see how important concepts are related to each other. All of the formulas are available to our subscribers, but here is a sample of the lesson you will see when you log in:



Lesson: 

Here are the two most important things to know about the relationship between differentiation and continuity:

  • When a function is continuous at some point it is not guaranteed to be differentiable at that point.
  • If a function is differentiable at a point then it is guaranteed to be continuous at that point.

If someone said to us that a continuous function is differentiable then all we would have to do is show one example where that was not true to prove them wrong. While there are many examples, the one that you need to keep in mind is that of the absolute value function <Sign in to see all the formulas>.

Let's look at the continuity and differentiability of <Sign in to see all the formulas> at $x=0$. First, we know that

<Sign in to see all the formulas>



Therefore, by the definition of continuity, <Sign in to see all the formulas> is continuous at $x=0$. (In fact it is continuous everywhere as the problem shows.) But what about its derivative? We have shown here that its derivative does not exist at $x=0$! This is because the function has a slope of +1 to the right of $x=0$ and -1 to the left of $x=0$. The slope to the left and to the right of a point must be the same for the function to be differentiable there. This is because the derivative at a point is a function limit and we know that for a function limit to exist the left-sided limit and the right-sided limit must equal.

So the function is continuous at $x=0$, but is not differentiable there! So the statement that a 'continuous function is differentiable' is not true in general.

Absolute value function demonstrating continuity and the lack of differentiability.

Finally, here is the proof that when a function is differentiable it is continuous.