Continuity is defined by limits.

*continuity*.

It is very important to note that saying

“a function is continuous at a point ”

is really making **three** statements:

- (a)
- is defined. That is, is in the domain of .
- (b)
- exists.
- (c)
- .

The first two of these statements are implied by the third statement, but are important enough that we want to break them up to highlight their significance.

Building from the definition of *continuity at a point*, we can now define what it
means for a function to be *continuous* on an open interval.

Loosely speaking, a function is continuous on an interval if you can draw the function on that interval without any breaks in the graph. This is often referred to as being able to draw the graph “without picking up your pencil.”

- Constant function
- is continuous on .
- Identity function
- is continuous on .
- Power function
- is continuous on .
- Exponential function
- is continuous on .
- Logarithmic function
- is continuous on .
- Sine and cosine
- Both and are continuous on .

In essence, we are saying that the functions listed above are continuous wherever they are defined, that is, on their natural domains.

### Left and right continuity

At this point we have a small problem. For functions such as , the natural domain is . This is not an open interval. What does it mean to say that is continuous at when is not defined for ? To get us out of this quagmire, we need a new definition:

Now we can say that a function is continuous at a left endpoint of an interval if it is right continuous there, and a function is continuous at the right endpoint of an interval if it is left continuous there. This allows us to talk about continuity on closed intervals.

**continuous on a closed interval**if is continuous on , right continuous at , and left continuous at ;**continuous on a half-closed interval**if is continuous on and right continuous at ;**continuous on a half-closed interval**if is is continuous on and left continuous at .