Problem: Investigate continuity of the following function:
Solution: As a combination of elementary functions, the formula with arctangent is continuous on its domain, which happens to be exactly the set on which it determines our function f. Thus we know that the given f is continuous on the open set
It remains to explore continuity at 0. This is done by checking on one-sided
limits at 0 and comparing the outcomes (if the limits converge) mutually and
with
Since the one-sided limits are equal, it follows that the function f
has a limit at 0 equal to
This numer has to be compared with f (0) = 1. Since these two numbers are different, f is not continuous at 0, and we see that it ha a removable discontinuity there.
Conclusion: f is continuous on