- Exams have been tentatively scheduled as follows:
- Monday, 2/15
- Monday, 3/22
- Monday, 4/19
- New assignment:
Tue 1/26 Section 2.4/2.5 - Read 2.7.
- Begin problem set 2.5 on IMathAS (due Thursday, 2/4, at midnight).
- Definition: continuity at a point
Otherwise
is discontinuous at
There are various kinds of discontinuity:
- Removable discontinuities
- Jump discontinuities
- Infinite discontinuities
- Maybe a better definition: a function is continuous on a domain if its graph can be traced there without lifting the pencil from the paper.
- Most of the functions we've looked at so far satisfy this, although not all: recall the function of 2.2, #2:
This function has a limit at zero (-.5), but is not defined there.
- Since continuity is wrapped up with limits, we have the same kinds
of results that we did in relation to limits:
- We define one-sided limits
- we have sum laws, product, and quotient laws.
- Classes of functions that are continuous on their domains:
- Polynomials
- Rational functions: (ratios of polynomials)
- Trigonometric functions: sine, cosine, tangent
- Root functions:
- Exponential functions:
- For all these functions one can evaluate limits using the
so-called "substitution method": simply plug in
for
.
- An important theorem: Theorem 4, p. 68, RE composite functions
- Roughly: A composition of continuous functions is continuous.
- Let
be a composite function. If
is continous at
and
, then
is continuous at
- Examples:
- #14, p. 71
- #20, p. 71
- This was based on a NASA
module:
-
-
- Bad Passwords
- What's the big idea? Indeterminate forms cause trouble.
Strategy: transform f(x) algebraically if possible into a new expression that is defined and continuous at x=c and then evaluate by substitution.
- Examples
- 1, p. 77
- 3, p. 77
- 20, p. 78
- 32, p 78