# Math Help - Continuous Function on a Closed Bounded Interval

1. ## Continuous Function on a Closed Bounded Interval

Assuming the theorem that a continuous real-valued function on a closed
bounded interval is bounded and attains its bounds, prove that if f : R → R is
continuous and f(x) → +∞ as x → ±∞ then there exists some x_0 ∈ R such that
f(x) > f(x_0) for all x ∈ R.

I'm not even sure where to start with this - doesn't anybody have any hints.

Thanks.

2. Hello,

Let $x_1$ be a real number such that for all $x\leqslant x_1$, $f(x)\geqslant f(42)$.
Let $x_2$ be a real number such that for all $x\geqslant x_2$, $f(x)\geqslant f(42)$.

Can you show that $x_0$ exists and that $x_0\in[x_1,x_2]$ ?