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Continuity
Curves without gaps?
Animation
(infinite length)
Continuity
Definition A function f is continuous
at a number a if
lim f(x) = f(a).
x -> a
1. f (a) is defined.
2. lim f(x) exists.
x -> a
3.lim f(x) = f(a).
x -> a
Continuity
1. f (a) is defined.
2. lim f(x) exists.
x -> a
3.lim f(x) = f(a).
x -> a
Animation
sin (1/x)
If f is not continuous at a , we say f is
discontinuous at a, or f has a
discontinuity at a .
Example Where is the function
f(x)=(x 2 – x – 2)/(x – 2) discontinuous?
Definition A function f is continuous
from the right at a number a if
lim f(x) = f(a),
x -> a +
and f is continuous from the left a if
lim f(x) = f(a).
x -> a -
Definition A function f is continuous
on an interval if it is continuous at
every number in the interval.
At an endpoint of the
interval we understand
continuous to mean
continuous from the
right or continuous from
the left.
Example Use the definition of
continuity and the properties of limits
to show that the function
_____
f (x) = x 16 –x2 is continuous on the
interval [-4, 4].
Theorem If f and g are continuous
at a and c is a constant, then the
following functions are also
continuous at a:
1. f + g
2. f – g
4. f g
5. ( f / g) if g(a) is not
3. c f
equal to 0.
Theorem
(a)Polynomials are continuous
everywhere; that is continuous on
R = (-, ).
(b)Any rational function is
continuous wherever it is defined;
that is, it is continuous on its
domain.
Theorem The following types of
functions are continuous at every
number in their domains:
-polynomials
-rational functions
-root functions
-exponential functions
-trigonometric functions
-inverse trigonometric functions
-logarithmic functions.
Example Evaluate
lim arctan
x -> 2
2
((x -
4) / (3x
2–
6x)).
Theorem If f is continuous at b and
lim g(x) = b, then, lim f(g(x)) = f(b).
x -> a
x -> a
In other words,
lim f(g(x)) = f(lim g(x)).
x -> a
x -> a
Theorem If g is continuous at a
and f is continuous at g(a), then
(f o g)(x) = f(g(x)) is continuous at a.
The Intermediate Value Theorem
Suppose that f is continuous on the
closed interval [a, b] and let N be
any number strictly between f (a)
and f (b). Then there exists a number
c in (a,b) such that f (c)= N .
Example Use the Intermediate Value
Theorem to show that there is a root of
the given equation in the specified
interval.
ln x = e –x , (1,2).