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  "body": "\n## Definition\n\nLet $I \\subseteq \\R$ be an [[interval|0Q1D85B8G6]].\nA [[function|0EPAT51DXQ]] $f \\colon I \\to \\R$ is *midpoint convex*, or *Jensen-convex*, if\n$$f\\left(\\frac{x + y}{2}\\right) \\le \\frac{f(x) + f(y)}{2} \\quad \\text{for all } x, y \\in I.$$\n\n## Remarks\n\nExample: $f(x) = x^2$ on $\\R$ is midpoint convex.\nCheck: $\\frac{x^2 + y^2}{2} - \\left(\\frac{x + y}{2}\\right)^2 = \\frac{(x - y)^2}{4} \\ge 0$.\n\nNon-example: $f(x) = -x^2$ on $\\R$ is not midpoint convex, since $f(0) = 0 \u003e -1 = \\frac{f(-1) + f(1)}{2}$.\n\nEvery [[convex function|07Q7CNM5WF]] is midpoint convex: take $t = 1/2$ in the definition of convexity.\nThe converse is false in general.\nA [[discontinuous additive function|FFTNS66XG6]] $a \\colon \\R \\to \\R$ satisfies $a\\big(\\frac{x+y}{2}\\big) = \\frac{a(x) + a(y)}{2}$, so it is midpoint convex.\nBut it is not convex, since convex functions on open intervals are continuous.\nA midpoint convex function that is continuous is convex.\n",
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