Limits | Continuous Functions on Open and Closed Intervals | 6 |

The continuity also depends upon the range of the function. The continuity of the function is checked on all values that are in the interval. Interval can either be closed or open. If open, then it means you need to check from the point after the starting interval point till the point which is before the ending interval point. However, if you are provided with a close interval then you have to check your function against the values that are in the interval which includes starting and ending point of the interval. A function f(x) is continuous at the open interval [a, b] if:

The function is right-continuous, i.e., \lim_{ x \rightarrow { a }^{ + } } f(x)
The function is left continuous, i.e., \lim_{ x \rightarrow { b }^{ - } } f(x)
If f is continuous at a closed interval [a, b], then f is bounded on that interval.

Thus
We say a function f is continuous on the open interval (a, b) if f is continuous at every point in (a, b). We say f is continuous on the closed interval [a, b] if f is continuous on (a, b), continuous from the right at a, and continuous form the left at b.