[Hand out time sheets]
Prof. Tibor Beke will substitute for me on Thursday and
Friday of this week.
Section 1.5: Continuity
Key concept:
Continuity.
A continuous function is one that satisfies the direct
substitution property lim
x
→
a
f
(
x
) =
f
(
a
).
Main points of the section?
..?.
.
Main point #1:
Most of the functions normally
encountered in precalculus math are continuous.
Main point #2:
Most ways of combining two continuous
functions give functions that are continuous too.
Main point #3:
If a function
f
is continuous, it satisfies
the intermediate value theorem.
Definition: We say the function
f
(
x
) is
continuous
at a
iff
(1) lim
x
→
a
f
(
x
) =
f
(
a
), i.e., in symbols,
f
(
x
)
→
f
(
a
) as
x
→
a
.
Otherwise, we say
f
(
x
) is
discontinuous at a
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentIf a function is continuous at every point in its domain, we
say that it is
everywhere continuous
, or (just)
continuous
.
Example 1: One of our limit laws tells us that (for all
values
a
) lim
x
→
a
x
2
=
a
2
, so the function
f
(
x
) =
x
2
is
continuous at
a
for every
a
; that is, it is continuous at every
point in its domain (in this case, the domain is
R
, the set of
all real numbers).
Example 2: The function sqrt(
x
), whose domain is [0,
∞
), is
not continuous at 0, because it is not defined throughout a
punctured interval {
x
: 0 < 
x
–0 <
r
}, no matter how small
r
is.
Example 3(a): The Heaviside function
H
(
x
) is not
continuous at 0 (though it is continuous at every other value
of
x
) because lim
x
→
0
H
(
x
) does not exist (more specfically,
the onesided limits lim
x
→
0+
H
(
x
) and lim
x
→
0–
H
(
x
) exist but
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '11
 Staff
 Calculus, Continuity, Intermediate Value Theorem, $1, Continuous function, Inverse function, L. Theorem

Click to edit the document details