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All of the elementary functions are continuous (because they are continuous at the x-values where they are defined.

Sometimes we want to talk about the limit of a function as x approaches infinity or negative infinity (∞ or - ∞). This is essentially the same idea: approaching ∞ means that x is getting larger and larger; approaching - ∞ means smaller and smaller.

Rigorous Definitions

We now make precise the intuitive definitions of limit and continuity given above. Let f be a function from some subset of the real numbers to the real numbers and let x0 be a real number. Then the function f is said to have limit L at x0 if for all ε > 0, there exists a δ > 0 such that 0 < | x - x0| < δ implies | f (x) - L| < ε. If this is the case, we write

f (x) = L    

As above, if a function f has a limit L = f (x0) at x0, then f is said to be continuous at x0. A function that is continuous at every point in its domain is said to be a continous function.

As an example of a proof that uses this definition, we show that the linear function f (x) = 3x is continuous at x0 = 1. Given ε > 0, we choose δ = ε/3. Suppose | x - 1| < δ. Then | f (x) - f (1)| = | 3x - 3| = 3| x - 1| < 3δ = ε. Therefore the limit of f (x) at x = 1 is f (1) = 3, and f is continuous there.

Intermediate Value Theorem

We conclude by mentioning an important property of continuous functions. Suppose f (x) is continuous on an interval [a, b]. Let y be any number between f (a) and f (b). Then the intermediate value theorem states that there exists c in the interval (a, b) such that f (c) = y.

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