# Continuity on an interval.

Definition 5.5.1   Let be a function defined on an open interval . The function is continuous on if it is continuous at every point of .

Example 5.5.2
• A polynomial function is continuous on .
• A rational function is continuous on every interval contained in its domain.
• The functions and are continuous on .
• The functions and are continuous on every interval contained in their respective domains. For instance, is continuous on .
• The logarithmic functions are continuous on .
• The exponential functions are continuous on ..

Example 5.5.3   Let be defined as follows:

On the interval , is the restriction of a polynomial function, thus it is continuous. In particular it is left continuous at .

On the interval , is the restriction of a polynomial function, thus it is continuous. Let us compute the left limit of at 1:

The function is right-continuous at 1 if, and only if , i.e. if, and only if, .

In conclusion, is continuous over the whole of whenever ; otherwise, it is continuous at every point of and has a discontinuity at .

Example 5.5.4   Let . We wish to study the continuity of . We have:

Part of the graph of is displayed on Figure 5.

On each interval of the form , the function coincides with the restriction of a linear function, thus it is continuous on each interval of this form. In particular is right-continuous at every integer . Let us compute the left limit of at an integer :

At the point , the value of is given by . Thus the following results hold:
• The function is left-continuous at 0 and right-continuous at 0, hence it is continuous at 0.
• If , the function is right-continuous at but is not left-continuous at 0.

Proposition 5.5.5   If is continuous on the interval , then is an interval.

Example 5.5.6 (see Figure 5)   Take . Then .

Proposition 5.5.7   If is continuous on the closed interval , then is a closed interval.

Example 5.5.8 (see Figure parab interv 1)   . Take . Then .

The importance of the continuity hypothesis appears with the function whose graph is displayed on Figure 5(b): there is a point of discontinuity at . Take the interval ; then and this is not an interval.

An important corollary of this proposition will come in the next subsection (i.e Thm intermediate value 1 and Thm 6.3).

Noah Dana-Picard 2007-12-28