Continuity

I.  Quiz

II.  Homework

III.   Continuity

Pictures will be presented on the board.

Definition:  A function is continuous at a x = c if the lim as x approaches c = f(x).

In other words, if the graph has no holes asymptopes, or ,breaks then the function is continuous.  Or if you can draw the function without lifting your pencil then it is continuous.  

Continuous Functions:  

1)  Polynomials

2)  sin and cos

3)  Rational Functions where the denominator is nonzero

4)  Sums, Differences, and Products of continuous functions

5)  Quotients of continuous functions

6)  Compositions of continuous functions

Examples:

The following are continuous:

A)  y = x² + 3x - 4

B)  y = x sin x

C)  y = 1/(1+ x²)

Exercises:  Determine whether the following are continuous.  If they are not continuous, at which points are they discontinuous?

A)  y = (x - 1)/(x + 1)

B)  y = x/(x² + 3x - 4)

C)  y = 2x + 3 for x < 1 and 3x - 2 for x > 1

D)  y = x²  for x < 2 and 5x - 6 for x > 2

For what value of k is the function continuous?

f(x) = 3x²  - 5 for x < 1 and 5x + k for x > 1

IV.  One Sided Limits

For a function with a break the limit does not exist, however it is still interesting to consider where the path is heading towards on the left side and where it heading on the right.  For example if

f(x) = |x|/x

then for x negative f(x) = -1  while for x positive, f(x) = 1.  We write

V.  The Intermediate Value Theorem

If f is continuous on [a,b]  and f(a) < k < f(b) then there exists at least one number c in the closed interval [a,b] for which f(c) = k

In particular if f(a) and f(b) have different signs, then f has a root between a and b.  

Exercise:  Write psudocode to find a root of a function