Convexity: Examples and Counterexamples

The function $f(x)=|x|$ is convex on $\mathbb R$. Indeed, for $0\le\lambda\le1$,

$|\lambda x+(1-\lambda)y|\le|\lambda x|+|(1-\lambda)y|=\lambda|x|+(1-\lambda)|y|,$

by the triangle inequality (The triangle inequality) and absolute homogeneity (Basic properties of the absolute value), which is precisely the convexity inequality (Convex, strictly convex, concave, strictly concave, and midpoint-convex real functions on an interval).

Convexity does not entail differentiability at every point; the nondifferentiability of absolute value at zero is recorded in $x \mapsto |x|$ is continuous everywhere and not differentiable at $0$: the difference quotient equals $1$ on the right and $-1$ on the left, so the two one-sided limits differ but is not a dependency of this example.

For nonnegative weights $\lambda_i$ with $\sum_i\lambda_i=1$ and real $x_i$,

$\left(\sum_i\lambda_i x_i\right)^2\le\sum_i\lambda_i x_i^2.$

The function $q(t)=t^2$ is convex because

$\lambda x^2+(1-\lambda)y^2-(\lambda x+(1-\lambda)y)^2=\lambda(1-\lambda)(x-y)^2\ge0$

for $0\le\lambda\le1$, using nonnegativity of squares (Squares of nonzero elements are positive). Applying finite Jensen (Finite Jensen inequality for a convex function and nonnegative weights summing to one) to $q$ gives the displayed inequality.

Every midpoint-convex function $\mathbb R\to\mathbb R$ is continuous, and hence convex.

For $f(x)=x^3$, the power rule gives $f''(x)=6x$ (For a natural $n \ge 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $\iota(n)\,x^{\,n-1}$; for $n = 0$ it is the constant $1$, with derivative $0$; for a natural $n \ge 1$ the function $x \mapsto x^{-n}$ is differentiable at every $x \ne 0$ with derivative $-\iota(n)\,x^{-n-1}$; consequently every polynomial function is differentiable at every real, with the derivative computed term by term). It is negative on $(-\infty,0)$ and positive on $(0,\infty)$, while $f$ is continuous at $0$. Therefore the sign-change criterion (A continuous function whose second derivative has opposite signs on the two sides of a point has an inflection point there) makes $0$ an inflection point (An inflection point as a point of continuity where convexity changes to concavity or conversely).