The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages.

Home Continuous Functions Continuity Theorem 4
 
del.icio.us Reddit Facebook StumbleUpon MySpace Twitter Search the VIAS Library  |  Index

Theorem 4

The next theorem is similar to the Chain Rule for derivatives.

THEOREM 4

If f is continuous at c and G is continuous at f(c), then the function

g(x) = G(f(x))

is also continuous at c.

That is, a continuous function of a continuous function is continuous.

PROOF

Let x be infinitely close to but not equal to c. Then

st(g(x)) = st(G(f(x))) = G(st(f(x))) = G(f(c)) = g(c).

For example, the following functions are continuous:

f(x) =03_continuous_functions-114.gif

all x

g(x) = |x3-x|,

all x

h(x) = (1 + √x)1/3,

x > 0

j(x) = esin x,

all x

k(x) = ln|x| ,

all x ≠ 0

Following are two examples illustrating two types of discontinuities:

Example 5
Example 6

Home Continuous Functions Continuity Theorem 4

Last Update: 2006-11-05