Continuity of Compositions


What about a function like

x + cos(ln(x))
f(x) =
?
1 - sqrt(x)

This is made up from functions in the Field Guide using the operation of composition. Here is a useful tool for dealing with compositions.

Composition Law: Suppose that f(x) = g(h(x)) for functions g and h. If h is continuous at x = a, and g is continuous at h(a), then f is continuous at x = a.

Why is THAT true?

Example: Where is the function cos(ln(x)) continuous?

Solution: The function ln(x) is defined and continuous for all positive x. The function cos(x) is continuous for all x. Hence by the composition law cos(ln(x)) is continuous for all positive x.


Now what about the example

x + cos(ln(x))
f(x) =
?
1 - sqrt(x)

The function f(x) = x is continuous for all real numbers and as we have seen the function cos(ln(x)) is continuous for all positive x, so x + cos(ln(x)) is continuous for all positive x . . . UH OH!

VROOOOOM...

Pork Chop will not let you get away with that! She pulls you over to the side of the trail and reaches into a well-worn black leather sack. Out comes a pamphlet which she presses it into your hand.

VROOOOOOOOM!!!

As she leaves in a black cloud of bad-smelling exhaust, you glance at the pamphlet. It is full of ridiculous looking diagrams printed on cheap newsprint. The title proudly proclaims:

"Limit Laws. They Work For All Of Us"


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All rights reserved---1996
William A. Bogley
Robby Robson