Convex and monotone functions are one-dimensional models of order, curvature, and optimization. They have strong regularity even when not differentiable.
Convexity by chords
Convexity means the graph lies below its chords.
Definition. A function \(f:I\to\mathbb R\) on an interval is convex if \(f((1-\lambda)x+\lambda y)\le(1-\lambda)f(x)+\lambda f(y)\) for \(0\le\lambda\le1\).
Proposition. For convex \(f\), slopes of secant lines are monotone: if \(x\lt y\lt z\), then \[ \frac{f(y)-f(x)}{y-x}\le \frac{f(z)-f(x)}{z-x}\le \frac{f(z)-f(y)}{z-y}. \]
Proof. Write \(y=(1-\lambda)x+\lambda z\). Convexity gives \(f(y)\le(1-\lambda)f(x)+\lambda f(z)\), which rearranges to the first inequality. Writing \(y\) between \(x\) and \(z\) and rearranging symmetrically gives the second.
One-sided derivatives
Convex functions have controlled directional slopes.
Theorem. A convex function on an open interval has finite left and right derivatives at every point, and these one-sided derivatives are increasing.
Proof. The monotonicity of secant slopes shows that the limits of \((f(x+h)-f(x))/h\) as \(h\downarrow0\) and \(h\uparrow0\) exist as monotone limits bounded by nearby secant slopes. Comparing secant slopes at different base points gives monotonicity of the one-sided derivatives.
Theorem (interior regularity). A finite convex function on an open interval is locally Lipschitz and therefore continuous. More precisely, if \[ a<u<v<b,\qquad [a,b]\subset I, \] then for all \(x,y\in[u,v]\), \[ |f(x)-f(y)| \le L|x-y|,\qquad L=\max\left\{\left|\frac{f(u)-f(a)}{u-a}\right|, \left|\frac{f(b)-f(v)}{b-v}\right|\right\}. \]
Proof. For \(u\le x<y\le v\), monotonicity of secant slopes gives \[ \frac{f(u)-f(a)}{u-a} \le\frac{f(y)-f(x)}{y-x} \le\frac{f(b)-f(v)}{b-v}. \] Taking absolute values yields the estimate. Hence one Lipschitz constant works on every compact subinterval of \(I\).
Jensen inequality
Convexity converts averages of inputs into bounds on averages of outputs.
Theorem. If \(f\) is convex and \(\lambda_i\ge0\), \(\sum_i\lambda_i=1\), then \[ f\left(\sum_i\lambda_i x_i\right)\le \sum_i\lambda_i f(x_i). \]
Proof. The case of two points is the definition. The finite case follows by induction: combine the first \(n-1\) points into their weighted average, then apply the two-point inequality to that average and \(x_n\).
Corollary (integral Jensen inequality). Let \((\Omega,\mathcal A,\mu)\) be a probability space, let \(X\) and \(f(X)\) be integrable, and let \(f\) be convex on an interval containing the range of \(X\). Then \[ f\!\left(\int_\Omega X\,\mathrm{d}\mu\right) \le\int_\Omega f(X)\,\mathrm{d}\mu. \]
Proof. At \(m=\int X\,\mathrm{d}\mu\), choose a supporting slope \(p\in[f'_-(m),f'_+(m)]\). Then \(f(x)\ge f(m)+p(x-m)\) for every \(x\). Integrating gives \[ \int f(X)\,\mathrm{d}\mu \ge f(m)+p\int(X-m)\,\mathrm{d}\mu=f(m). \] This argument also shows why existence of a supporting line is the geometric heart of Jensen's inequality.
Monotone functions
Monotone functions can jump, but only in a controlled way.
Theorem. A monotone real function on an interval has left and right limits at every point and has at most countably many discontinuities.
Proof. For increasing \(f\), the left limit is \(\sup_{y\lt x}f(y)\) and the right limit is \(\inf_{y>x}f(y)\). A discontinuity has a positive jump interval \((f(x-),f(x+))\). For each \(m\), the jumps of size at least \(1/m\) are finite on bounded subintervals, because their disjoint jump intervals fit inside the total variation. Taking a countable union over \(m\) and bounded intervals gives countability.
Subgradients
Subgradients replace derivatives at corners.
Definition. A number \(p\) is a subgradient of a convex \(f\) at \(x\) if \(f(y)\ge f(x)+p(y-x)\) for all \(y\).
Proposition. At an interior point \(x\), \[ \partial f(x)=[f'_-(x),f'_+(x)]. \] Consequently, \(x\) minimizes \(f\) if and only if \(0\in\partial f(x)\).
Proof. If \(p\in\partial f(x)\), division of \[ f(y)-f(x)\ge p(y-x) \] by \(y-x\), separately for \(y<x\) and \(y>x\), gives \(f'_-(x)\le p\le f'_+(x)\). Conversely, any \(p\) in this interval lies below every right secant slope and above every left secant slope by monotonicity of slopes, so the supporting inequality holds for all \(y\). Finally, \(0\in\partial f(x)\) is exactly \(f(y)\ge f(x)\) for all \(y\).
Remark. These one-dimensional facts are the prototypes for subdifferentials, supporting hyperplanes, and optimality conditions in Optimization.