Example:
What kind of discontinuity does the function f(x) = x sin(1/x)
have at x = 0 ?
f(x) = x sin(1/x)
Since | x sin(1/x) | < | x |, we can see that the
limit of f(x) as x approaches zero from either
side is zero. Hence, the function has a removable
discontinuity at zero. If we set f(0) = 0 then
f(x) is continuous.