# Arithmetic operations for continuity

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Let’s discuss some methods to create a new continuous function from known continuous functions.

Continuity of sums, products, quotients and compositions of continuous functions

Let $$a$$ be a real number.

1. Suppose $$f$$ and $$g$$ are functions continuous in $$a$$. Then, the functions $$f + g$$ and $$f \cdot g$$ are also continuous in $$a$$. If $$g(a)\ne0$$, then we have the same for $$\frac{f}{g}$$.
2. Suppose $$f$$ is a function in $$a$$ and that $$g$$ is a function continuous in $$f(a)$$. Then the composition $${g}\circ{f}$$ is continuous in $$a$$.

Continuity of power functions

Let $$d$$ be a real number. If $$f$$ is a function that is continuous in $$a$$ with $$f(a)>0$$, then $$f(x)^d$$ is also continuous in $$a$$.

Let’s practice!

Consider the function rules $$f(x)=x^2$$ and $$g(x)=x+1$$.

What is the function rule of the composition $$f\circ g$$?

$$\begin{array}{rcl} f \circ g(x) &=& f \left(g(x) \right) \\ &&\phantom{xyzuvw}\color{blue}{\text{composition of functions}} \\ &=& g(x)^2 \\ &&\phantom{xyzuvw}\color{blue}{\text{function rule of }f\text{ entered}} \\ &=&\left(x+1\right)^2 \\ &&\phantom{xyzuvw}\color{blue}{\text{function rule of }g\text{ entered}} \\ \end{array}$$

$$f\circ g(x) = \left(x+1\right)^2$$

Let’s practice once more!

Let $$f$$ be the function with rule $$f(x) = {{1}\over{x^2+4}}$$ and $$g$$ the function with rule $$g(x)=x^2$$.

Determine the function rule for $${f – g}$$.

$$\left({f – g}\right)(x)=f(x)-g(x)=\left({{1}\over{x^2+4}}\right)-\left(x^2\right)={{-x^4-4\cdot x^2+1}\over{x^2+4}}$$

$$\left({f – g}\right)(x)\,=\, {{-x^4-4\cdot x^2+1}\over{x^2+4}}$$