# Continuity & Properties of functions

1. May 7, 2008

### ricardianequiva

1. The problem statement, all variables and given/known data

Let f:R->R be a continuous function where limit as x goes to positive/negative infinity is negative infinity. Prove that f has a maximum value on R.

2. Relevant equations
None

3. The attempt at a solution
I tried to use the definition of infinite limits but I'm not sure how to do this explicitly.

2. May 7, 2008

### HallsofIvy

Let X be a real number. Since $\lim_{x\rightarrow \infty}= -\infty$, there exist x1[/sup] such that if x> x1, f(x)< X. Since $\lim_{x\rightarrow -\infty}= -\infty$, there exist x2 such that if x< x2, f(x)< X. Now look at the values of f(x) on the closed and bounded interval [x1, x2].

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