You should be a high school student, right? Different fields have different interpretations of smooth curves. The definition of calculus in advanced mathematics is that if the function f(x) has a first-order continuous derivative in the interval (a, b), then its graph is a curve with tangents everywhere, and the tangents rotate continuously with the movement of the tangents. Such a curve is called a smooth curve.

The words of high school students can be understood as the existence of tangents at every point on the curve. Not all curves have tangents, but smooth curves have tangents at every point. This involves the definition of curve. The curves you came into contact with in high school are all smooth, so in your opinion, all points have tangents. You'll know later.

The movement of the tangent point keeps turning. That is, the tangent point changes slowly, and the tangent slope gradually becomes larger or smaller. For example, the function of the square of x is on the right side of 0, starting from 0, and the tangent slope is 0. The farther to the left, the greater the slope and the greater the angle. This is rotation.

If you are a college student, I can give you an example. f(x)=x^2*sin( 1/x),f(0)=0。

F is differentiable everywhere, but the derivative is discontinuous at 0. In other words, the curve (x, f(x)) is not smooth at the origin.