1. 1.4
Continuity
and
One-Sided Limits
This
will test
the
“Limits”
of your
brain!

2. Definition of Continuity
A function is called continuous at c if the
following three conditions are met:
1. f(c) is defined 2. 3.
A function is continuous on an open
interval (a,b) if it is continuous at each
point in the interval.

3. Two Types of Discontinuities
Removable
Point Discontinuity
Non-removable
Jump and Infinite
the open circle
can be filled in
to make it
continuous
Removable example

4. Non-removable discontinuity.
Ex. -1 1

5. Determine whether the following functions are
continuous on the given interval.
yes, it is
continuous
( ) 1

6. ( )
discontinuous at x = 1
removable discontinuity since filling in (1,2)
would make it continuous.

7. yes, it is continuous

8. One-sided Limits
Limit from the right
Limit from the left
Find the following limits
D.N.E. 1
D.N.E.

9. Step Functions “Jump”
Greatest Integer -1
D.N.E.

10. g(x)=
Is g(x) continuous at x = 2? 3 3
g(x) is continuous at x = 2

11. Intermediate Value Theorem
If f is continuous on [a,b] and k is any number
between f(a) and f(b), then there is at least one
number c in [a,b] such that f(c) = k
f(a)
In this case, how
many c’s are there
where f(c) = k? k
f(b) 3
[ ]
a b

12. Show that f(x) = x3 + 2x –1
has a zero on [0,1].
f(0) = (0) – 1 = -1
f(1) = (1) – 1 = 2
Since f(0) < 0 and f(1) > 0, there must
be a zero (x-intercept) between [0,1].