For functions that are "normal" enough, we know immediately whether or not they are continuous at a given point. Nevertheless, the continuity of a function is such an important property that we need a precise definition of continuity at a point:

That is, f is continuous at c if and only if for all e > 0 there exists a d > 0 such that

In words, for x close to c, f(x) should be close to f(c).

• If f is continuous at every real number c, then f is said to be continuous.
• If f is not continuous at c, then f is said to be discontinuous at c. The function f can be discontinuous for two distinct reasons:
  • f(x) does not have a limit as x® c. (Specifically, if the left- and right-hand limits exist (and are finite) but are different, the discontinuity is called a jump discontinuity.)
  • f(x) has a limit as x® c, but lim_{x® c} f(x) ¹ f(c) or f(c) is undefined. (This is called a removable discontinuity, since we can ``remove'' the discontinuity at c by redefining f(c) as lim_{x® c} f(x).)

Rather than returning to the e-d definition whenever we want to prove a function is continuous at a point, we build up our collection of continuous functions by combining functions we know are continuous:

If f and g are continuous at c, then

1. f+g is continuous at c.
2. af is continuous at c for any real number a.
3. fg is continous at c.
4. f/g is continuous at c if g(c) ¹ 0.

Example

The function f(x) = (x²-4)/[(x-2)(x-1)] is continuous everywhere except at x = 2 and at x = 1. The discontinuity at x = 2 is removable, since (x²-4)/[(x-2)(x-1)] can be simplified to (x+2)/(x-1). To remove the discontinuity, define

We can also look at the composition f o g of two functions,

If g is continuous at c and f is continuous at g(c), then the composition f o g is continuous at c.

We'd also like to speak of continuity on a closed interval [a,b]. To deal with the endpoints a and b, we define one-sided continuity:

A function f is continuous from the left at c if and only if lim_{x® c^-} f(x) = f(c). It is continuous from the right at c if and only if lim_{x® c⁺} f(x) = f(c).

We say that f is continuous on [a,b] if and only if

1. f is continuous on (a,b),
2. f is continuous from the right at a, and
3. f is continuous from the left at b.

Note that f is continuous at c if and only if the right- and left-hand limits exist and both equal f(c).

Example

The function

f(x) = ì ï ï ï í ï ï ï î x, x £ 0 x2, 0 < x £ 1 2 x , 1 < x £ 2 x-1, x > 2

That is, f is continuous at c if and only if for all e > 0 there exists a d > 0 such that

In words, for x close to c, f(x) should be close to f(c).