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.
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
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.
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?
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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First, let's talk about the-- all differentiable functions are continuous relationship. Think about it for a moment. If a function is differentiable, then it has a slope at all points of its graph. If this is so, then the graph is a single continuous curve that you can draw with one pen stroke. But if the function is not differentiable, then it may have a gap in the graph, like we have in our blue graph.
Continuous with no Differentiable Relationship
Now think about the not all continuous functions are differentiable relationship. A function is continuous if it has no gaps, so the function of the absolute value of x is a continuous function because the function doesn't break up. There are no gaps. But is this function differentiable? Does it have a slope at all points of the graph? It might seem like it at first, but look at the second graph here at the point x = 0. Can you say that it has a slope? Think about what would happen if you were to try to ski over that point? Would you have a slope to ski on or would you tip over? You would tip over. This is because this function has no slope at that point. This function, although being continuous, is no differentiable.
We can specify the domain where the function is differentiable though. We can say that the absolute value of x is differentiable for all points, except x = 0. But the function as a whole is not differentiable? Wasn't that interesting?
Now let's review what you've learned. A continuous function is a function whose graph is a single unbroken curve. A discontinuous function then is a function that isn't continuous. A function is differentiable if it has a derivative. You can think of a derivative of a function as its slope.
The relationship between continuous functions and differentiability is-- all differentiable functions are continuous but not all continuous functions are differentiable.
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