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).)
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
- f+g is continuous at c.
- αf is continuous at c for any real number
α.
- fg is continous at c.
- 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
We can also look at the composition f o g of two functions g and f,
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 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
- f is continuous on (a,b),
- f is continuous from the right at a, and
- 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
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].