Convex functions

Beta · **Joined:** Thu, 30 Oct 2008 10:47:09 UTC **Posts:** 182

1. If f is convex on $[0,\infty)$ , then $\lim_{x\to\infty}\frac{f(x)}{x}$ exists.

2. If $f:R^n\to R$ is convex, then there exists k>0 such that $\liminf_{|x|\to\infty}\frac{f(x)}{|x|}\geq-k.$

outermeasure · **Posted:** Wed, 29 Jun 2011 17:04:36 UTC

Beta wrote:

1. If f is convex on $[0,\infty)$ , then $\lim_{x\to\infty}\frac{f(x)}{x}$ exists.

2. If $f:R^n\to R$ is convex, then there exists k>0 such that $\liminf_{|x|\to\infty}\frac{f(x)}{|x|}\geq-k.$

1. The limit only exists in $\overline{\mathbb{R}}$ , not $\mathbb{R}$ , e.g. f(x)=x^2

is convex and $\dfrac{f(x)}{x}=x\to+\infty$ . Slopes of support lines is nondecreasing, hence limit.

2. Similar to 1.

S.O.S. Mathematics CyberBoard

Convex functions

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