How Continuous Functions Stay Smooth

Continuous functions stay smooth when they are added, subtracted, multiplied by a constant, or multiplied together. If f and g are continuous at a point a, then f+g, f-g, cf, and fg are also continuous at a. This makes it possible to build more complex functions from simpler continuous parts without breaking continuity.

A function is continuous over a domain if its graph is a smooth, unbroken curve without gaps or sudden jumps.

Continuous functions can be combined using algebraic operations, and the result will usually remain continuous.

For example, x squared plus one is continuous across all real numbers, as there are no gaps or sudden jumps. Simi...