Continuity

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:

A function f is continuous at c if and only if limx→ c f(x) = f(c).

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

if  |x-c| < δ   then |f(x)-f(c)| < ε.

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

Notes

  • 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 limx→ c f(x) ≠ f(c). (This is called a removable discontinuity, since we can ``remove'' the discontinuity at c by redefining f(c) as limx→ c f(x).)

Figures

Rather than returning to the ε-δ 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. αf is continuous at c for any real number α.
  3. fg is continous at c.
  4. f/g is continuous at c if g(c) ≠ 0.

Example

The function f(x) = (x2-4)/((x-2)(x-1)) is continuous everywhere, but at x = 2 and at x = 1 f(x) is indefined. We can define f(2) in such a way that f(x) is continuous at x=2 since (x2-4)/((x-2)(x-1)) can be simplified to (x+2)/(x-1). We define

f(2) = 2+2
2-1
= 4.

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

(f o g)(x) = f(g(x)).

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

Proof

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 limx→ c - f(x) = f(c). It is continuous from the right at c if and only if limx→ 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.

Figures

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) =
















x,
x ≤ 0
x2,
0 < x ≤ 1
2
x
,
1 < x ≤ 2
x-1,
x > 2
is continuous everywhere except at x = 1, where f has a jump discontinuity.

It can be proved that
  if f is continuous on an interval I then the range f(I) of f is an interval.

i.e. if a and b belong to I, for every real number D strictly between f(a) and f(b) there is an element c of (a,b) such that f(c) = D [Intermediate Value Theorem].
In particular if f(a) and f(b) have different signs, there is an element c of (a,b) such that f(c) = 0 (c is said a zero of f).
And it can be proved that
  if f is continuous on an closed interval [a,b] then f has a maximum at some point in [a,b] and a minimum at some point in [a,b
[Extreme Value Theorem].


Key Concepts [index]

A function f is continuous at c if and only if limx→ c f(x) = f(c).

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

if  |x-c| < δ   then |f(x)-f(c)| < ε.

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