query: if there exists a discontinuous function f(x) which satisfies $f(\frac{x+y}{2})\leqslant\frac{1}{2}f(x)+\frac{1}{2}f(y)$ but is not convex?
positive_passages.docid: 20614
positive_passages.text: Example of a function such that $\varphi\left(\frac{x+y}{2}\right)\leq \frac{\varphi(x)+\varphi(y)}{2}$ but $\varphi$ is not convex
negative_passages.docid: 7358
negative_passages.text: For each $y \in \mathbb{R}$ either no $x$ with $f(x) = y$ or two such values of $x$. Show that $f$ is discontinuous.
negative_passages.docid: 55679
negative_passages.text: Does there exist a continuous function from [0,1] to R that has uncountably many local maxima?
negative_passages.docid: 65827
negative_passages.text: When $\min_{x \in X,y \in Y} f(x,y) = \min_{x \in X} \min_{y \in Y} f(x,y)$?
negative_passages.docid: 85153
negative_passages.text: Show that $f(x) = -\ln(x)$ is convex (WITHOUT using second derivative!)
negative_passages.docid: 20688
negative_passages.text: Discontinuous functions with closed graphs
negative_passages.docid: 140445
negative_passages.text: Sequence of continuous functions whose pointwise limit is discontinuous
negative_passages.docid: 16463
negative_passages.text: Is the function $\mbox{tr}(XAX')$ convex?
negative_passages.docid: 4679
negative_passages.text: Nonpiecewise Function Defined at a Point but Not Continuous There
negative_passages.docid: 11167
negative_passages.text: A counterexample of a differentiable function on an interval whose derivative is discontinuous
negative_passages.docid: 28392
negative_passages.text: Construct a monotone function which has countably many discontinuities
negative_passages.docid: 116382
negative_passages.text: Showing convexity of a function with the restriction over an arbitrary line
negative_passages.docid: 80210
negative_passages.text: It is posible to have a function $f$ with the following property?
negative_passages.docid: 126931
negative_passages.text: Find $f(x)$ such that $f(x)+f\left(\frac{1}{x}\right)=f(x)\cdot f\left(\frac{1}{x}\right)$
negative_passages.docid: 24810
negative_passages.text: Functions satisfying $(b-a)f'(\tfrac{a+b}{2}) = f(b)- f(a)$
negative_passages.docid: 6636
negative_passages.text: Is there a positive function $f$ on real line such that $f(x)f(y)\le|x-y|, \forall x\in \mathbb Q , \forall y \in \mathbb R \setminus \mathbb Q$?
negative_passages.docid: 35671
negative_passages.text: Why can a discontinuous function not be differentiable?