1. Existence of the function at the point:
Before anything else, it's essential to determine whether the function is defined at the specific point under consideration.
2. Evaluating the limit of the function as it approaches the point:
Calculate the limit of the function as it approaches the given point from both sides to ensure that it converges to a finite value.
3. Comparing the function's value to the limit at the point:
Compare the actual value of the function at the point with the limit. If they match, it indicates continuity.
4. Investigating left-hand and right-hand limits:
Examine the behavior of the function as it approaches the point from the left and right sides to detect any discontinuities.
5. Checking for removable discontinuities:
Look for points where the function might have discontinuities that can be removed by redefining the function at that point.
6. Identifying jump discontinuities:
Jump discontinuities occur when the function "jumps" from one value to another at a specific point.
7. Examining infinite discontinuities:
Infinite discontinuities arise when the limit of the function at the point approaches positive or negative infinity.
8. Analyzing oscillatory behavior:
Some functions exhibit oscillatory behavior, causing them to oscillate infinitely close to a point. This may indicate a discontinuity.
9. Verifying continuity through graphical representations:
Plotting the function graphically can provide visual confirmation of continuity or the presence of discontinuities.
10. Utilizing the epsilon-delta definition of continuity:
For a rigorous approach, the epsilon-delta definition can be employed to establish continuity mathematically.
