Limits | Continuity of Polynomials, Rationals and Composition | 7 |

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There are many examples of continuous functions. For example, consider a polynomial function
p(x). We know the domain of
p(x) is the set of all reals. This in combination with one of our limit laws, " whenever p(x)p(x) is a polynomial function," tells us that limx→cp(x) and
p(x) both exist and agree in value for every real number cc. Thus, all polynomial functions are continuous everywhere (i.e., at any real value c).

With arguments like these (ones that appeal back to the limit laws for simple functions and combinations of functions), we can similarly deduce the following functions are continuous on their domains:

polynomial functions

rational functions (i.e., quotients of two polynomials)

Additionally, if a function gg is continuous atx=c and a function f is continuous at x=g(c), then the compositionf(g(x)) is continuous at c.