Check whether the function is defined at the point of interest
Compute the limit of the function as the input approaches that point
Evaluate the function at that point
Compare the limit and the function value
If the limit exists and equals the function value, the function is continuous at that point
If the limit does not exist, the function is not continuous at that point
If the limit exists but does not equal the function value, the function is not continuous at that point
Check continuity on intervals by verifying continuity at every point in the interval
Use continuity rules for sums, products, quotients, and compositions of continuous functions
Watch for common discontinuities such as holes, jumps, vertical asymptotes, and undefined points