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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 $$x_0$$.

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