/sci/ - Science & Math

I've started reading Spivak as I haven't done calculus in years and I am going back to school for my masters. This example has me confused.

There I get the example and reasoning pic related.
>Theorem 3:
>If f is continuous on [a, b], then there is some number y in [a, b] such that
f(y) >= f(x) for all x in [a, b].
There is no y in [0, 1] because x can be a number so y<x<1.


My question is, why is the function f(x) = 1/x for x = (0, infinity) regarded as continuous but pic related not?
What's stopping you saying for all y in (0, infinity) you can chose an x so that 0<x<y and f(y) does not bound f(x)?