left derivative/right derivative 307

This numerical value represents the proportion in

total variation in the yvariable that can be accounted

for by the line of best fit. This proportion has values

between 0 and 1, with a value of 1 indicating that all

variation is due to a linear fit, that is, the data values

lie perfectly on the regression line, and an R2value of

zero indicates that none of the variation in the y-val-

ues is due to a linear correlation. If R2= 0.84, for

example, then we can say that 84 percent of the vari-

ation in the y-values is accounted for by a linear rela-

tionship with the values of x.

See also

REGRESSION

.

Lebesgue, Henri-Léon (1875–1941) French Analy-

sis Born on June 28, 1875, French mathematician

Henri Lebesgue is remembered for his revolutionary

ideas in

CALCULUS

and in the theory of integration. By

generalizing the notion of

AREA

to one of an abstract

“measure theory,” Lebesgue transformed the object of

an integral into a tool applicable to an extraordinarily

large class of settings. He published work on this topic

at the young age of 27.

Lebesgue studied at the École Normale Supérieure,

France, and taught at the University of Nancy for 3

years. He presented his famous work to his university

colleagues during his final year there in 1902. The

idea behind his approach is relatively simple. One typ-

ically computes an integral by subdividing the range

of inputs, the x-axis, into small intervals and then

adding the areas of rectangles above these intervals of

heights given by the function. This is akin to counting

the value of a pocket full of coins by taking one coin

out at a time and adding the outcomes as one goes

along. Lebesgue’s approach, however, is to subdivide

the range of outputs, the y-axis, into small intervals

and to measure the size of the sets on the x-axis for

which the function gives the desired output on the y-

axis. This is akin to counting coins by first collecting

all the pennies and determining their number, then all

the nickels and ascertaining the size of that collection,

and so forth. Of course the shape of the sets one

encounters along the x-axis can be complicated and

difficult to measure in size. The work of French math-

ematicians Émile Borel (1871–1956) and M

ARIE

E

NNEMOND

C

AMILLE

J

ORDAN

(1838–1922) in devel-

oping so-called measure theory provided Lebesgue the

means to do this.

Lebesgue wrote over 50 papers and two books,

including his 1902 paper “Intégrale, longeur, aire”

(Integrals, lengths, area), his 1910 article “Sur l’intégra-

tion des functions discontinues” (On the integration of

discontinuous functions), and his 1906 monograph

Leçons sur les séries trigonométriques (Lectures on

trigonometric series). He also made important contribu-

tions to the fields of

TOPOLOGY

and F

OURIER SERIES

,

and was appointed professor at the Sorbonne, Univer-

sity of Paris, in 1910.

At one point in his life, Lebesgue expressed serious

unease about continuing the work in integration theory

he himself had founded. He feared that by making

mathematics abstract, topics in the subject would begin

to lose meaningful context. He died in Paris, France, on

July 26, 1941.

Lebesgue’s revolutionary approach to integration

theory is taught to all upper-level college students and

graduate students in mathematics today. It is consid-

ered a core component of any serious study of analysis.

left derivative/right derivative The

DERIVATIVE

of a

function f(x) at position xis defined as the

LIMIT

f′(x) =

. The value of the limit, if it exists,

represents the slope of the tangent line to the graph of

the function at position x. If the quantity his restricted

to run only through negative values as it approaches

the value zero, that is, if the limit above is replaced by

a limit from the left, we obtain the left derivative of the

function at x:

Restricting hto run only through positive values pro-

duces the right derivative of the function at x:

(See

LIMIT

.) The general derivative f′(x) exists if, and

only if, the left and right derivatives both exist and agree

in value. This need not always be the case. Consider the