Editing Convex function (section)

=== Functions of one variable ===

* Suppose <math>f</math> is a function of one [[real number|real]] variable defined on an interval, and let <math display=block>R(x_1, x_2) = \frac{f(x_2) - f(x_1)}{x_2 - x_1}</math> (note that <math>R(x_1, x_2)</math> is the slope of the purple line in the first drawing; the function <math>R</math> is [[Symmetric function|symmetric]] in <math>(x_1, x_2),</math> means that <math>R</math> does not change by exchanging <math>x_1</math> and <math>x_2</math>). <math>f</math> is convex if and only if <math>R(x_1, x_2)</math> is [[monotonically non-decreasing]] in <math>x_1,</math> for every fixed <math>x_2</math> (or vice versa). This characterization of convexity is quite useful to prove the following results.
* A convex function <math>f</math> of one real variable defined on some [[open interval]] <math>C</math> is [[Continuous function|continuous]] on <math>C </math>. Moreover, <math>f</math> admits [[Semi-differentiability|left and right derivatives]], and these are [[monotonically non-decreasing]]. In addition, the left derivative is left-continuous and the right-derivative is right-continuous. As a consequence, <math>f</math> is [[differentiable function|differentiable]] at all but at most [[countable|countably many]] points, the set on which <math>f</math> is not differentiable can however still be dense. If <math>C</math> is closed, then <math>f</math> may fail to be continuous at the endpoints of <math>C</math> (an example is shown in the [[#Examples|examples section]]).
* A [[differentiable function|differentiable]] function of one variable is convex on an interval if and only if its [[derivative]] is [[monotonically non-decreasing]] on that interval. If a function is differentiable and convex then it is also [[continuously differentiable]].
* A differentiable function of one variable is convex on an interval if and only if its graph lies above all of its [[tangent]]s:<ref name="boyd">{{cite book| title=Convex Optimization| first1=Stephen P.|last1=Boyd |first2=Lieven| last2=Vandenberghe | year = 2004 |publisher=Cambridge University Press| isbn=978-0-521-83378-3| url= https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf#page=83 |format=pdf | access-date=October 15, 2011}}</ref>{{rp|69}} <math display=block>f(x) \geq f(y) + f'(y) (x-y)</math> for all <math>x</math> and <math>y</math> in the interval. 
* A twice differentiable function of one variable is convex on an interval if and only if its [[second derivative]] is non-negative there; this gives a practical test for convexity. Visually, a twice differentiable convex function "curves up", without any bends the other way ([[inflection point]]s). If its second derivative is positive at all points then the function is strictly convex, but the [[converse (logic)|converse]] does not hold. For example, the second derivative of <math>f(x) = x^4</math> is <math>f''(x) = 12x^{2}</math>, which is zero for <math>x = 0,</math> but <math>x^4</math> is strictly convex.
**This property and the above property in terms of "...its derivative is monotonically non-decreasing..." are not equal since if <math>f''</math> is non-negative on an interval <math>X</math> then <math>f'</math> is monotonically non-decreasing on <math>X</math> while its converse is not true, for example, <math>f'</math> is monotonically non-decreasing on <math>X</math> while its derivative <math>f''</math> is not defined at some points on <math>X</math>.
* If <math>f</math> is a convex function of one real variable, and <math>f(0)\le 0</math>, then <math>f</math> is [[Superadditivity|superadditive]] on the [[positive reals]], that is <math>f(a + b) \geq f(a) + f(b)</math> for positive real numbers <math>a</math> and <math>b</math>.
{{math proof|proof=
Since <math>f</math> is convex, by using one of the convex function definitions above and letting <math>x_2 = 0,</math> it follows that for all real <math>0 \leq t \leq 1,</math>  <math display=block>
\begin{align}
f(tx_1) & = f(t x_1 + (1-t) \cdot 0) \\
    & \leq t f(x_1) + (1-t) f(0) \\
    & \leq t f(x_1). \\
\end{align}
</math> From <math>f(tx_1)\leq t f(x_1)</math>, it follows that <math display=block>
\begin{align}
f(a) + f(b) & = f \left((a+b) \frac{a}{a+b} \right) + f \left((a+b) \frac{b}{a+b} \right) \\
& \leq \frac{a}{a+b} f(a+b) + \frac{b}{a+b} f(a+b) \\
& = f(a+b).\\
\end{align}</math> 
Namely, <math>f(a) + f(b) \leq f(a+b)</math>.
}}

* A function <math>f</math> is midpoint convex on an interval <math>C</math> if for all <math>x_1, x_2 \in C</math> <math display=block>f\!\left(\frac{x_1 + x_2}{2}\right) \leq \frac{f(x_1) + f(x_2)}{2}.</math> This condition is only slightly weaker than convexity. For example, a real-valued [[Lebesgue measurable function]] that is midpoint-convex is convex: this is a theorem of [[Wacław Sierpiński|Sierpiński]].<ref>{{cite book|last=Donoghue|first=William F.| title= Distributions and Fourier Transforms|year=1969|publisher=Academic Press | isbn=9780122206504 |url= https://books.google.com/books?id=P30Y7daiGvQC&pg=PA12|access-date=August 29, 2012|page=12}}</ref> In particular, a continuous function that is midpoint convex will be convex.