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My notes do not include a proof to the following:

If $p$ is a polynomial, then $p$ is continuous at every point in $R$. If $r = \frac{p}{q}$ is a ratio of two polynomials, then it is continuous at every point in $R$ where $q \not = 0$.

Can I prove this through the Sequential Continuity type theorem wherein I say that $p$ tends to a limit within delta and so the combination of it and $q$ also converges by the algebraic properties of limits?

Any help would be appreciated, thanks!

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    What do you know about the continuity of sums, products and ratios of continuous functions? – 2017-01-19
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    @dxiv they are all also continuous, and thus any polynomial p is, including the $p/q$ I am asking about I suppose? – 2017-01-19
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    If $f(x)$ is continuous then $1/f(x)$ is continuous at points where $f(x) \ne 0\,$ so $p/q = p \cdot 1/q$ will be continuous at points where $q \ne 0$. – 2017-01-19

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