1.5 Continuity
Most of the techniques of calculus require that functions
be continuous. A function is continuous if you can draw it
in one motion without picking up your pencil.
Definition: A function f is continuous at a number a if
lim f ( x)  f (a)
xa
(the limit is the same as the value of the function).
This function is discontinuous
(has discontinuities) at x=1 and
x=2.
2
1
1
2
3
4
It is continuous everywhere else
on the interval [0,4].
Removable Discontinuities:
(You can fill the hole.)
Essential Discontinuities:
jump
infinite
oscillating
• 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 at a number 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.
Examples on the board.
• 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
3. cf
f
4. fg
5.
if g (a)  0
g
• Theorem: The following types of functions are continuous
at every number in their domains: polynomials, rational
functions, root functions, trigonometric functions.
Example: The function
x3
2 x
x2
is continuous on the intervals [0,2) and (2,)
Intermediate Value Theorem
If a function is continuous between a and b, then it takes
on every value between f  a  and f  b  .
f b
Because the function is
continuous, it must take on
every y value between f  a 
and f  b  .
f a
a
b
Examples on the board.