Continuity Over an Interval
Now that we have explored the concept of continuity at a point, let’s extend it to continuity over an interval. A function is continuous over an interval if you can trace it without lifting your pencil between any two points within that interval.
- Open Interval: A function is continuous on an open interval if it is continuous at every point within that interval.
- Closed Interval: A function is continuous on a closed interval if it is continuous on , continuous from the right at , and continuous from the left at .
continuity over an interval
- A function is continuous over an open interval if it is continuous at every point in the interval.
- A function is continuous over a closed interval of the form if it is continuous at every point in and is continuous from the right at and is continuous from the left at .
- Analogously, a function is continuous over an interval of the form if it is continuous over and is continuous from the left at .
Requiring that and ensures that we can trace the graph of the function from the point to the point without lifting the pencil. If, for example, , we would need to lift our pencil to jump from to the graph of the rest of the function over .
How To: Determine Continuity Over an Interval
- Check Continuity on Open Interval: Verify the function is continuous at all points within .
- Check Right Continuity at : Ensure
- Check Left Continuity at : Ensure .
State the interval(s) over which the function is continuous.
State the interval(s) over which the function is continuous.