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Functions, Limits, Continuity



Summary Continuity

A function is considered continuous if it is continuous at all points in its domain.

Some Important Continuous Functions

You might recognize that the formal requirement of continuity, i.e. that

f (x) = f (c)    

is a property of polynomial functions. Thus, all polynomial functions are continuous. The following functions are always continuous, and you should be aware of them:
1. Polynomial Functions
2. Rational Functions, wherever the denominator is nonzero.
3. sin(x) and cos(x)
4. The sum, difference, product, and quotient (as long as the denominator is nonzero) of two continuous functions is continuous.

Demonstrating the Continuity of a Piecewise Function

One problem that you might have to deal with is using the formal definition of continuity to determine whether a piecewise-defined function is continuous.

Example: is f a continuous function?

f (x) =    


For a function to be continuous, it must be continuous at every point in its domain. The obvious point for us to be worried about here is the point where the definition of f changes, i.e. at x = 2. At places other than at x = 2, f is defined by polynomial functions, which we know are continuous. It is the point where these two continuous functions meet that concerns us.

Therefore, to prove that f is a continuous function, we must prove that it is continuous at x = 2. In other words, we must show that