Understanding Continuity and Smoothness in Functions

You know a function is continuous if you can draw its graph without ever lifting your pen.

Think of a smooth, flowing line—no breaks, no jumps, no holes.

That's a continuous function. But the second you have to lift your pen to get from

one point to another, you've found a discontinuity. It's literally a break in the graph. But here’s the twist.

A function can be perfectly continuous, with no gaps, yet still not be "smooth" everywhere.

Imagine the sharp V-shape of the absolute value graph. You can draw it in one stroke,

but that sharp point means it's not differentiable there.