Continuously differentiable

Okay kiddo, imagine we have a line that goes up and down, kind of like a roller coaster. When we say it's "differentiable", it means that we can find the slope of the line at any point, like how steep it is going up or down.

But sometimes, the line isn't just smooth like a roller coaster - there might be bumps or twists or turns. If we can find the slope of the line at every single point on it, even if it's all bumpy, we call that "continuously differentiable". It's like knowing exactly how steep the roller coaster is no matter where you are on it, even if it's going up and down and all around.

And just like how it's easier to ride a smooth roller coaster, it's easier to work with a continuously differentiable line in math.