Draw the graph of a function. Roughly speaking, if there’s no holes, jumps, or other choppy weirdness it’s a continuous function. The function is connected to itself like a curvy rope laid out on the ground, with no cuts.
Now if that function has no sharp points, it’s a differentiable function. Again roughly speaking, imagine that the rope is free of kinks or sharp bends. The absolute value function is an example of a function which is not differentiable at the origin: there’s a sharp point there.
It’s well known that if a function is differentiable, it’s also continuous. This is formally provable in a couple of lines, and it makes sense. If a function is smooth, it can’t possibly be disconnected from itself. But what’s a little less obvious is that the converse is not true. A function can be continuous and not differentiable.
Now in a way this is not implausible. The absolute value function isn’t differentiable at the origin because of that sharp point, but it is connected to itself there and thus it’s continuous. But that’s just a problem at one point – the rest of the function is both continuous and differentiable. What’s weird is that there are functions which are continuous everywhere and differentiable nowhere. Here’s one due to Weierstrass:
Image credit Mathworld
It’s continuous everywhere. It’s not differentiable over any finite interval. Zoom in between two sharp points, and you’ll find more small sharp points. Keep zooming and you’ll keep finding even smaller sharp points between those. Almost every single point on this function is sharp, there’s nowhere the function is smooth for any length at all. I say almost because there is a disjoint set of points of measure zero which are differentiable, but there’s no interval however small in which every point is differentiable.
Fortunately these are mathematical curiosities and do not appear in any physical situation I’ve ever heard of – though you never know.
Mathematics. Lewis Carrol couldn’t make this stuff up.