How do I find the continuity or discontinuity of a graphed function? I am stuck on this problem and cannot whatsoever get past it.
What problem?
To determine the continuity or discontinuity of a graphed function, you can follow these steps:
1. Look for any breaks, holes, or jumps in the graph. These are potential points of discontinuity.
2. Check if the function is defined at these potential points of discontinuity. If the function is not defined at a particular point, it is a point of discontinuity. For example, if there is a hole in the graph where the function is not defined, it is a removable discontinuity.
3. If the function is defined at a point, check if the one-sided limits exist. This means evaluating the function as it approaches the point from the left side (using a value slightly smaller than the point) and from the right side (using a value slightly larger than the point).
i. If the left-hand and right-hand limits are equal, the function has a removable discontinuity or a point of continuity.
ii. If the left-hand and right-hand limits are different, the function has a jump discontinuity.
4. Finally, check if the function approaches infinity (∞) or negative infinity (-∞) as it goes towards a particular point. If the function tends to infinity, it is an infinite discontinuity.
By following these steps, you can systematically analyze a graphed function and determine its continuity or discontinuity. Remember to consider both the points where the function is not defined and the behavior of the function as it approaches these points.