an analysis problem?

Question

for any p>0, always exists d>0, such that |f(x)-f(1)|>=p whenever |x-1|>=d, this means:
(a) f(x) is continuous at x=1
(b) f(x) is discontinuous at x=1
(c) f(x) is unbounded
(d) lim f(x)=infinite as |x| goes to infinite
which choice is right?

Answer 1

It's clearly not a or b -- this is about what happens away from x=1, not near x=1.

c is clearly correct -- if B were a bound for F, let p = 2*max(B, 100) and we contradict B being a bound in a hurry.

d is NOT correct, because the function can be nasty and unbounded in some particular region, and then calm down again as x goes to infinity. E.g., f(x) could equal tan(x) for x <= 1300, but equal sin(x) for x >1300