query: How to prove that $f$ is convex function if $f(\frac{x+y}2)\leq \frac12f(x) + \frac12f(y)$ and $f$ is continuous?
positive_passages.docid: 16943
positive_passages.text: Midpoint-Convex and Continuous Implies Convex
negative_passages.docid: 109942
negative_passages.text: Find continuous functions such that $f(x+y)+f(x-y)=2[f(x)+f(y)]$
negative_passages.docid: 129856
negative_passages.text: Show that $f(x)=\frac{x}{1+|x|}$ is uniformly continuous.
negative_passages.docid: 15491
negative_passages.text: Prove that $f:\mathbb R^n\to\mathbb R$ is affine if and only if it is convex and concave
negative_passages.docid: 20192
negative_passages.text: Prove that: $(f(x))^{2} + (f'(x))^{2} \leqslant \max(a,b)$ where $(a,b) \in \mathbb{R}^2$
negative_passages.docid: 145368
negative_passages.text: How to show $f:Y\to X^{+}$ is continuous?
negative_passages.docid: 116548
negative_passages.text: How to prove the real value function $f$ is convex on $[a,d]$ if $f$ is convex on both $[a,c]$ and $[b,d]$? ($a<b<c<d$).
negative_passages.docid: 56081
negative_passages.text: Prove that the ellipsoid $E = \{ (x,y,z) \in R^3 \mid \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} \leq 1 \}$ is convex
negative_passages.docid: 92029
negative_passages.text: Is this rational function convex?
negative_passages.docid: 118628
negative_passages.text: Is the function $f(x)=|x|^{1/2}$ Lipschitz continuous?
negative_passages.docid: 54764
negative_passages.text: Given a real function $g$ satisfying certain conditions, can we construct a convex $h$ with $h \le g$?
negative_passages.docid: 40905
negative_passages.text: Prove $f(\sum^k_{i=1} \alpha_i x_i) \leq \sum^k_{i=1} \alpha_i f(x_i) $ for a convex function f
negative_passages.docid: 63526
negative_passages.text: Proving that $(\sup_{x\in R}|f'(x)|)^2\leq 4\sup_{x\in R}|f(x)|\cdot\sup_{x\in R}|f''(x)|$.
negative_passages.docid: 85153
negative_passages.text: Show that $f(x) = -\ln(x)$ is convex (WITHOUT using second derivative!)
negative_passages.docid: 86305
negative_passages.text: A secant inequality for convex functions
negative_passages.docid: 55866
negative_passages.text: Convex + Monotone =? Convex
negative_passages.docid: 71619
negative_passages.text: Proving that a convex function is Lipschitz