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In calculus, we work with functions that are continuous. In fact, we usually require functions that are also differentiable, meaning that at every point, the function can be approximated with a tangent line. If a function is differentiable at a point, does that mean it is continuous at that point?

Some functions, like are differentiable everywhere. Click on the image below, and slide the point of tangency to see what this looks like.

What about Is it continuous everywhere?

Using desmos, have a look at Is it continuous? Where is it not differentiable?

Can there be a function that is continuous everywhere, but is not differentiable at any point? Let's investigate this function:

Now, start zooming in using the desmos controls. You can fine-tune the zoom using the settings:

What happens? Do you think this function has a tangent at any point?