Question

Use limit rules and the continuity of power functions to prove that every polynomial function is continuous everywhere.

Short answer

The polynomial function $\mathit{f}$ is continuous at$\mathit{x}\mathbf{=}\mathit{c}$.

Step-by-step solution

Step 1. Given Information:

Using limit rules and the continuity of power functions

Step 2. Prove:

Consider any polynomial function,

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}{\mathit{a}}_{\mathbf{n}}{\mathit{x}}^{\mathbf{n}}\mathbf{+}{\mathit{a}}_{\mathbf{n}\mathbf{-}\mathbf{1}}{\mathit{x}}^{\mathbf{n}\mathbf{-}\mathbf{1}}\mathbf{+}\mathbf{\cdots }\mathbf{+}{\mathit{a}}_{\mathbf{1}}\mathit{x}\mathbf{+}{\mathit{a}}_{\mathbf{0}}$

Assume that c is a real number.

Now by the sum rule, constant multiple rules, and limit of a constant, we have:

$\underset{\mathbf{x}\mathbf{\to }\mathbf{c}}{\mathbf{l}\mathbf{i}\mathbf{m}}\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}{\mathbf{a}}_{\mathbf{n}}\underset{\mathbf{x}\mathbf{\to }\mathbf{c}}{\mathbf{l}\mathbf{i}\mathbf{m}}{\mathit{x}}^{\mathbf{n}}\mathbf{+}{\mathbf{a}}_{\mathbf{n}\mathbf{-}\mathbf{1}}\underset{\mathbf{x}\mathbf{\to }\mathbf{c}}{\mathbf{l}\mathbf{i}\mathbf{m}}{\mathit{x}}^{\mathbf{n}\mathbf{-}\mathbf{1}}\mathbf{+}\mathbf{\cdots }\mathbf{+}{\mathbf{a}}_{\mathbf{1}}\underset{\mathbf{x}\mathbf{\to }\mathbf{c}}{\mathbf{l}\mathbf{i}\mathbf{m}}\mathit{x}\mathbf{+}{\mathit{a}}_{\mathbf{0}}$

Step 3. Every polynomial function is continuous everywhere:

Since power functions with positive integer powers are continuous everywhere, we can solve each of the component limits by evaluation, which gives us

$$\begin{gathered} \underset{\mathbf{x}\mathbf{\to }\mathbf{c}}{\mathbf{lim}}\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}{\mathit{a}}_{\mathbf{n}}{\mathit{c}}^{\mathbf{n}}\mathbf{+}{\mathit{a}}_{\mathbf{n}\mathbf{-}\mathbf{1}}{\mathit{c}}^{\mathbf{n}\mathbf{-}\mathbf{1}}\mathbf{+}\mathbf{\cdots }\mathbf{+}{\mathit{a}}_{\mathbf{1}}\mathit{c}\mathbf{+}{\mathit{a}}_{\mathbf{0}} \\ \mathbf{=}\mathit{f}\mathbf{\left(}\mathit{c}\mathbf{\right)} \end{gathered}$$

Therefore, the polynomial function $\mathit{f}$ is continuous at x = c.