query: If $f$ is continuous and $f(\frac{x+y}{2})\leq \frac{f(x)+f(y)}2$ for all $x , y \in (a,b)$, prove that $f$ is convex.
positive_passages.docid: 144001
positive_passages.text: If $f$ is continuous and $\,f\big(\frac{1}2(x+y)\big) \le \frac{1}{2}\big(\,f(x)+f(y)\big)$, then $f$ is convex
negative_passages.docid: 30423
negative_passages.text: interior points and convexity
negative_passages.docid: 29679
negative_passages.text: Prove: $|x-y|\leq |x|+|y|$
negative_passages.docid: 110112
negative_passages.text: Proving a function is Lipschitz continuous
negative_passages.docid: 36371
negative_passages.text: Exercise on convex functions
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: 40438
negative_passages.text: Proving that closed (and open) balls are convex
negative_passages.docid: 78834
negative_passages.text: if f(x) if differentiable and continuous, prove $\frac{af(a)-bf(b)}{a-b} = f(c) + cf'(c) $
negative_passages.docid: 82198
negative_passages.text: Proving Convexity of Multivariate Function using Conditional Univariate Convexity
negative_passages.docid: 64905
negative_passages.text: Let $f \in C [a,b]$ such that $\forall x\exists y$ such that $|f(y)| \leq \frac{1}{2}|f(x)|$. Prove that $\exists c\in [a,b]$ such that $f(c) = 0$.
negative_passages.docid: 35857
negative_passages.text: Characterization of log-convexity
negative_passages.docid: 60887
negative_passages.text: $d_A(x) = \inf_{a \in A} \|x-a\|$ is convex if and only if $A$ is convex
negative_passages.docid: 135867
negative_passages.text: $f(x)=x^3$ is convex nor concave without to derivate
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: 74950
negative_passages.text: Show a function for which $f(x + y) = f(x) + f(y) $ is continuous at zero if and only if it is continuous on $\mathbb R$
negative_passages.docid: 102959
negative_passages.text: Show that $f$ is convex if and only if $f\left( \sum_{i=1}^m\lambda_ix_i \right) \leq \sum_{i=1}^m\lambda_if(x_i)$
negative_passages.docid: 77217
negative_passages.text: Proof that a coordinate-wise convex function is convex?