# Thread: Proof: If a function f is convex on (a,b) then f is bounded below on (a,b)

1. ## Proof: If a function f is convex on (a,b) then f is bounded below on (a,b)

Hi,

i need to prove that , but i dont really know well convex function

anyone can help me to solve that prove

Prove that if a function f is convex on (a,b) then f is bounded below on (a,b)

2. Originally Posted by neset44
Prove that if a function f is convex on (a,b) then f is bounded below on (a,b)
Choose two points c, d with $\displaystyle a<c<d<b$, and let L be the straight line through the points $\displaystyle (c,f(c))$ and $\displaystyle (d,f(d))$. The convexity of f ensures that the graph of f lies above the line L in the intervals $\displaystyle (a,c)$ and $\displaystyle (d,b)$. Also, convex functions are continuous, so f is bounded below by continuity in the closed interval $\displaystyle [c,d]$. Those facts together imply that f is bounded below throughout the interval $\displaystyle (a,b)$.

3. thanx for help : )