1.4 Continuity and one-sided limits

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