# The algebra of continuous functions.

Theorem 5.2.1   If and are two continuous functions at , then is continuous at .

For example, take and as follows:

The functions and are disontinuous at 0, but is continuous at 0 ( is constant on ).

Theorem 5.2.2   If and are two continuous functions at , then is continuous at .

For example, take and as follows:

The functions and are discontinuous at 0, but is continuous at 0 ( is constant on ).

Using the fact that a constant function is continuous on its domain, Theorems 2.1 and 2.2 imply that the set of all the continuous functions on the interval is a real vector space.

Theorem 5.2.3   If and are two continuous functions at , and if , then is continuous at .

The same functions as above yield a counter-example to the converse: the function is constant and is equal to over , thus it is continuous at every point.

Theorem 5.2.4   If is continuous at and if is continuous at , then is continuous at .

For example, take a function defined over some open interval , with at least one discontinuity at, say, ; now take a function , constant over . Then is constant over , thus is continuous at every point.

Noah Dana-Picard 2007-12-28