Say I have a continuous function $f(x)$ on the open interval $(0, 1)$. There is no limit on how large $|f(x)|$ can be apart from the fact that it can't be infinity, it will always be a real number.
Now say I have a continuous function $f(x)$ on the closed interval $[0, 1]$. This too has no limit on how large $|f(x)|$ can be apart from the fact that it can't be infinity, it will always be a real number.
So it seems like there is no difference between them...and they are both unbounded? But from what I've read continuous functions on closed intervals are bounded...
This seems like a chicken and an egg scenario to me. Can anyone set me straight on the difference between continuous functions on open intervals vs. closed intervals?