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 \(f\circ g\)?

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

Answer:

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


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 \({f – g}\).

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

Answer:

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

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