1.4 Continuity and one- sided limits "Mathematics -- the subtle fine art." -- Jamie Byrnie Shaw
Feb 12, 2016
1.4 Continuity and one-sided limits
"Mathematics -- the subtle fine art." -- Jamie Byrnie Shaw
Objective
To determine the continuity of functions
To find one-sided limits
Top 10 excuses for not doing your math homework
#10. Galileo didn't know calculus; what do I need it for?
#9. "A math addict stole my homework. #8. I'm taking physics and the homework
in there seemed to involve math, so I thought I could just do that instead.
#7. I have the proof, but there isn't room to write it in the margin.
#6. I have a solar powered calculator and it was cloudy.
Cont… #5. I was watching the World Series and got tied up trying
to prove that it converged. #4. I could only get arbitrarily close to my textbook. I
couldn't actually reach it. (I reached half way, and then half of that, and then ...)
#3. I couldn't figure out whether i am the square root of negative one or i is the square root of negative one.
#2. It was Einstein's birthday and pi day and we had this big celebration! (This only works for March 14)
#1. I accidentally divided by zero and my paper burst into flames.
Intuitive approach
Formal definition
A function is continuous at c if:
1. f(c) is defined ( f(c) exists )
2.
3.
existsxfcx
)(lim
)()(lim cfxfcx
Simply put..
A function is continuous if you can draw it without picking up your pencil
Can be continuous over open or closed intervals, or the entire function
Examples
When are the following functions continuous?
x1
112
xx
0,1
0,1)(
2 xx
xxxf
Discontinuities
If a function is discontinuous at a point, the discontinuity may be removable or non-removable depending upon whether the limit of the function exists at the point of discontinuity
In other words: Removable – holes Non-removable- breaks or asymptotes
Removable discontinuities
Denominators of fractions that factor with the numerators
Ex
Holes occur where you can take the limit but the actual value does not exist
242
xx
1)45( 2
xxx
Non-removable discontinuities
Asymptotes- when the denominator is zero
Ex:
This is when the actual value and the expected value or limit do not exist
32x 4
22
xx
Properties of continuity
If b is a real number, and f and g are continuous at x = c, then…
1. Scalar multiple: bf(x) is continuous at c
2. Sum and difference: f + g is continuous at c
3. Product: fg is continuous at c 4. Quotient: f/g is continuous at c,
as long as g(c) does not equal 0
Left and right limits
We can look at limits from just the left or just the right
Right: Left:
)(lim xfcx
)(lim xfcx
Greatest integer function
Graph
Find:
xxf )(
)(lim2
xfx
)(lim2
xfx
Remember…
Three things must happen for a limit to exist
The limit from the right exists The limit from the left exists The limit from the right equals the
limit from the left
Intermediate Value Theorem (IVT)
If f is continuous on [a,b] and k is a number between f(a) and f(b) then there is a c such that if a < c< b then f(c)= k