# Arithmetic preserves continuity

Let \( X \) be a subset of \( \mathbb{R} \), let \( f : X \to \mathbb{R} \) and \( g : X \to \mathbb{R} \) be functions, and let \( x_0 \) be an element of \( X \). Then, if \( f \) and \( g \) are continuous at \( x_0 \), then so too are the functions \( f+g \), \( f-g \), \( \max(f,g) \), \( \min(f,g) \) and \( fg \). If \( g \) is non-zero on \( X \), then \( f/g \) is also continuous at \( x_0 \).

### Derivation

This proposition comes about by considering the limit laws for functions with the definition of continuity.

### Importance

This proposition can be used to prove the continuity of many functions. For example, given the continuity of the identiy function, \( f(x) = x \), at every point in \( \mathbb{R} \), we can show that the function \( g(x) := \frac{max(x^3 + 4x^2 + x + 5, x^4 - x^3)}{(x^2 - 4)} \) is continous at every point in \( \mathbb{R} \) except for two points \( x = +2, x = -2 \).

Below are some more examples. Utilizing the below propositions we can show the continuity of complicated functions like \( h(x):= \frac{|x^2 -8x + 7|^{\sqrt{2}}}{(x^2 + 1)} \).

#### Exponentiation is continuous, I

Let \( a > 0 \) be a positive real number. Then the function \( f : \mathbb{R} \to \mathbb{R} \) given by \( f(x):= a^x \) is continuous.

#### Exponentiation is continuous, II

Let \( p \) be a real number. Then the function \( f : (0, \infty) \to \mathbb{R} \) given by \( f(x):= x^p \) is continuous.

#### Absolute value is continuous

The function \( f : \mathbb{R} \to \mathbb{R} \) given by \( f(x) = |x| \) is continuous.

This follows from the continuity of \( max(x, -x) \).

#### Composition preserves continuity

Let \( X \) and \( Y \) be subsets of \( \mathbb{R} \), and let \( f: X \to Y \) and \( g: Y \to R \) be functions. Let \( x_0 \) be a point in \( X \). If \( f \) is continuous at \( x_0 \) and \( g \) is continuous at \( f(x_0) \), then the composition \( g \circ f : X \to \mathbb{R} \) is continuous at \( x_0 \).