Non-analytic smooth function

Okay kiddo, imagine you have a piece of paper with a line drawn on it. Now, if you zoom in very closely to the line, you will see that it looks like a bunch of tiny little dots or points.

Now, let's say we have a function, which is just a fancy way of saying a rule that tells us what to do with numbers. We can use this function to draw a line on our paper.

A smooth function means that if you zoom in really closely to the line, it still looks like a smooth curve without any bumps or jumps. An example of a smooth function would be a simple line equation like y = mx + b (where m and b are just numbers).

However, a non-analytic smooth function is a bit different. You see, there are some functions out there that are smooth, but they can't be described by a simple equation like y = mx + b. Instead, they need a more complicated equation that involves something called infinite sums or series.

These non-analytic smooth functions have funny little bumps and curves when you zoom in really close, so they don't look like straight lines or curves anymore. But they are still considered smooth because they don't have any sudden jumps or breaks in them. It's like a rollercoaster that has lots of twists and turns, but no sudden stops or drops.

So, in summary, a non-analytic smooth function is a rule that can draw a line or a curve on a piece of paper, but it needs a more complicated equation than a simple straight line. These functions have little bumps and wiggles when you zoom in closely, but they are still smooth and don't have sudden jumps or breaks in them.