Like the fire in a flame, curves are what we perceive. You can break down a curve into points, the instantaneous rate of change, f of x) dx, the derivative in the calculus, but this does not mean that the curve does not exist.

If you had met my wife you would not ask that.?

Mathematics is exact and abstract. That's why it's so useful in describing physics, economics and other complicated systems. A curve is a well defined path in some mathematical space. It is not found in nature. There can be approximates in the physical world that are modeled adequately by the concept of a curve. For example, I can model the position of an object vs time as a curve. This is usually a good model in everyday life although both space and time are discrete.

Do lines exist?

"There is no such thing as a linear relationship in real life."

• My Engineering 101 Prof, at the start of the first class -

If you date people that don't miss leg days at the gym then yes curves are real and a lot of fun.

Curves exist. They can be defined my mathematical formulas. [reviewcivilpe.com]

You're right, they don't exist in nature. In order for a curve to exist in the physical sense it would have to be made out of something. We don't measure curves, so they don't exist as far as we know. Maybe string theory will turn out to be true then one could argue that strings are curves.
But still the concept of a curve is extremely useful to us and it helps us make better predictions about the world, so not existing is not the worst a thing can be

To the eye and in the larger expanse of the universe yes. At the micro level, maybe not, just like a block wall is mostly empty space, but punch it and see how your hand feels.

Welcome to the pixelated universe ?

It seems to me that curves are either an abstraction, or they exist, and if they exist, that would mean other mathematical abstractions should exist too.

Have you ever looked at the path of a river?
Have you ever seen a snake in real life?
Have you ever watched ocean waves cresting?
Have you ever seen seashells?
Have you seen a chart representation of Pluto's orbit around our sun?
Etc.

You're welcome.

Haven't you seen any curvy women ?