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A function f: (a, b) → R is said to be convex if f(λx + (1 − λ)y) ≤ λf(x) + (1 − λ)f(y) whenever a <x, y <b and 0 <λ <1. Prove that every convex function is continuous. Usually it uses the fact: If a <s <t <u <b then f(t) − f(s) t − s ≤ f(u) − f(s) u − s ≤ f(u) − f(t) u − t.
Oct 19, 2018 · No: all convex functions f: R2 → R are continuous. Here's a slightly more general statement. Let f: Rn → R be a convex function, and let x ∗ ∈ Rn. We show that f is continuous at x ∗. Let S = {y ∈ Rn: ‖x ∗ − y‖ = 1}. Our first goal is to show that there's some M ∈ R such that f(y) ≤ M for all y ∈ S.
Let \(I\) be an interval and let \(f, g: I \rightarrow \mathbb{R}\) be convex functions. Prove that \(cf\), \(f + g\), and \(\max \{f, g\}\) are convex functions on \(I\), where \(c \geq 0\) is a constant. Find two convex functions \(f\) and \(g\) on an interval \(I\) such that \(f \cdot g\) is not convex. Answer. Add texts here.
Aug 16, 2019 · What you gave is the standard definition of a convex function. If $f$ is supposed to be continuous, it is enough to check that $$f\left(\frac{x+y}{2}\right) \le \frac{f(x)+f(y)}{2}$$ for all $x,y$. If $f$ is twice differentiable, it is enough to check that the second derivative is non negative.
Convex Functions. You are probably familiar with the notion of concavity of functions. Given a twice-di erentiable function ': R ! R, We say that ' is convex (or concave up) if '00(x) 0 for all x 2 R. We say that ' is concave (or concave down) if '00(x) For example, a quadratic function. '(x) = ax2 + bx + c. 0 for all x 2 R.
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Feb 9, 2018 · The idea is to use the condition of convexity to ‘squeeze’ the graph of f f near x x, as is shown in Figure 2. Figure 2: Given the values of f f in x x and on ∂P ={y1,y2} ∂ P = {y 1, y 2}, the convexity condition restricts the graph of f f to the grey area. For λ ∈[0,1] λ ∈ [0, 1] and y ∈∂P y ∈ ∂ P, the convexity of f f ...
5 days ago · A convex function is a continuous function whose value at the midpoint of every interval in its domain does not exceed the arithmetic mean of its values at the ends of the interval.
