- 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 ..

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 .

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.

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