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 , and let L be the straight line through the points and . The convexity of f ensures that the graph of f lies above the line L in the intervals and . Also, convex functions are continuous, so f is bounded below by continuity in the closed interval . Those facts together imply that f is bounded below throughout the interval .