- Your first exam is this Friday. You need
- Pencil or pen
- Scientific (non-graphing) calculator
- Your 1.6 homework is returned. Graded 8, 16, and 34.
- #8: don't be a limit dropper
- #16: don't be a limit dropper
- #34: don't be a limit dropper
- #8: straightforward -- but cite properties! You need to justify, and show work.
- #16: need to do some algebra (factoring)
- #34: "Use a graph...."; "Use a table..."; "Use the limit laws...." You really do need to do it all!
- The exam will formally cover only sections 1.4-1.6.
- However, you will also be expected to be prepared to handle problems like the secant and tangent line, using point-slope form, finding approximate roots, etc.
- Section 1.8: Continuity
- What is the difference between continuous and discontinuous things?
- A new administration in government?
- A sidewalk?
- Coffee poured into a cup?
- Definition: continuity at the point $x=a$:
\[
\lim_{x\rightarrow{a}}f(x)=f(a)
\]
Note: three things have to happen:
- The limit of $f$ exists at $a$,
- The function value $f(a)$ exists at $a$, and
- The two values are equal.
Otherwise $f$ is discontinuous at $a$.
There are various kinds of discontinuity (which we've already seen):
- Removable discontinuities (e.g. holes)
- Jump discontinuities
- Infinite discontinuities
- Crazy discontinuities like the function we discussed last time, which was $x^2$ for rational values of $x$, and 0 for irrational ones.
- Maybe a better definition:
A function is continuous on an interval if its graph can be traced there without lifting the pencil from the paper. (Try one! Make one smooth, and one really "jaggy".) - Many of the functions we've considered so far satisfy this, although not all do: consider the "holy function"
This function has a limit at zero (-.5), but is not defined there. If $f$ is not defined at $x=0$, then it cannot be continuous there. We can fix this, by the way.... Just define $f(0)$ to be the limit (the closed dot in the graph above).
- 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 continuity
- we have sum laws, product, and quotient laws. We've
already used these to demonstrate that polynomials are
continuous on all real numbers:
- Classes of functions that are continuous on their domains:
- Polynomials
- Rational functions: (ratios of polynomials)
- Trigonometric functions: sine, cosine, tangent
- Root functions: $f(x)=x^{1/n}$
- Exponential functions: $f(x)=b^x$
- For all these functions one can evaluate limits using the
so-called "substitution method": simply plug in $a$ for
$x$.
- An important theorem (p. 88), RE composite functions (remember that I told you how important compositions are? here's one place they show up....):
- Roughly: A composition of continuous functions is
continuous.
- Let $F(x)=f(g(x))$
be a composite function.
If $g$ is continous at $x=c$ and $f$ is continous at $g(c)$ then
$F(x)$ is continuous at $x=c$.
- Roughly: A composition of continuous functions is
continuous.
- Examples:
- pp. 90-93, #3, 10, 12, 19, 23, 36, 46 (with Mathematica)
- That crazy function which was $x^2$ for rational values of $x$, and 0 for irrational ones, has an interesting property: it is continuous at only a single point! It satisfies the definition only at $x=0$: it's discontinuous everywhere else! That's pretty weird.
- The Intermediate Value Theorem is an important result of continuity:
- Theorem: Suppose that f is continuous on the closed interval [a,b] and let N be any number between f(a) and f(b), where $f(a)\ne{f(b)}$. Then there exists a number $c$ in $(a,b)$ such that $f(c)=N$.
- Example: pp. 93, #69
- Theorem: Suppose that f is continuous on the closed interval [a,b] and let N be any number between f(a) and f(b), where $f(a)\ne{f(b)}$. Then there exists a number $c$ in $(a,b)$ such that $f(c)=N$.
- What is the difference between continuous and discontinuous things?