The Relationship Between Continuity & Differentiability

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  • 00:00 Functions
  • 00:22 Continuous vs. Discontinuous
  • 00:43 Differentiability
  • 1:38 The Relationship
  • 3:15 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Why is it that all differentiable functions are continuous but not all continuous functions are differentiable? Learn why in this video lesson. Also see what a continuous function looks like versus one that isn't.

Functions

There are two types of functions; continuous and discontinuous. A continuous function is a function whose graph is a single unbroken curve. You can draw the graph of any continuous function by using a single pen stroke without lifting your pen. A discontinuous function then, is a function that isn't continuous.

Continuous vs. Discontinuous

Continuous vs Discontinuous Graph
Continuous vs Discontinuous Graph

Look at these two functions pictured; can you tell which one is continuous and which one isn't? Is it the one with the red graph or the one with the blue graph? If you said that the red graph is the one that is continuous, and the blue graph, discontinuous, then you are absolutely right. Do you see how the blue graph is split into two parts? You can't draw this graph without lifting your pen.

Differentiability

A function is differentiable if it has a derivative. Functions can be differentiable as a whole or they can be differentiable only at certain locations. You can think of the derivative of a function as its slope, just like our mountains have slopes, so our functions have similar slopes. Just like the slope of a mountain tells us how steep it is, so our slope of our function tells us how steep the function is at a particular point.

We differentiate functions by following our differentiation rules. For example:

X^2 differentiates to 2x, while sin x differentiates to cosign x

Now, looking at our two graphs from earlier, do you see how one function has a slope throughout, while the other does not? The red graph has a slope throughout, while the blue graph doesn't have a slope at x = 0. What does this tell us about the relationship between continuous functions and differentiability?

The Relationship

We actually have a very interesting relationship between continuous functions and differentiability. The relationship is-- all differentiable functions are continuous, but not all continuous functions are differentiable. Why is this?

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