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Math and science::Analysis::Tao::09. Continuous functions on R

# 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 $$x_0$$.

p230-231