### 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.