In another thread, I explained that it is easy to show that the function defined by f(x) = x, if x is rational; 0, if x is irrational, approaches 0 at 0 using epsilon-delta. However, I am stuck in trying to show that f does not approach any number at a if a =/= 0 using the epsilon-delta definition for limits (NOT the sequential definition, which I will learn soon). Assuming the fact that the rationals and irrationals are dense in the reals, it's easy to show that f does not approach any number near a for the function f(x) = 1, if x is rational; 0 if x is irrational. But when x is allowed to vary, I'm not sure which epsilon I should be choosing. I always end up with two sets of inequalities that do not seem to lead to a contradiction. Any hints are appreciated.