For any given function,if anything like continuity,discontinuity,maxima,minima are asked, it means that we have to consider in its domain.
here domain = R-{0}.
Hence function is not defined at x = 0 at all. So we can't answer any question related to x = 0;
But yes when x->0, function is defined and its limiting value exists and it equals to '1'.
At x -> 0, functional gives maximum value also.
But at x=0; nothing can be said.
So if we defined function at x= 0 as f(0) = 1,then we can say function is continuous & maximum value exists at x=0.
So don't worry,IIT will never ask such confusing questions..just move on.