494 DERIVATES, EXTREMES, VARIATION 
that 2) take on define the upper and lower derivatives of as 
in I, 363. 
In a similar manner we introduce the upper and lower right- 
arid left-hand differential coefficients and derivatives, 
Rf , Rf , Lf , Lf. (3 
Thus, for example, 
Rf (a) = R ImT^ a+ h ] , 
h=o h 
finite or infinite. Cf. I, 336 seq. 
If f(x) is defined only in 51 = (a < /3), the points a, a + h must 
lie in 2i. Thus there is no upper or lower right-hand differential 
coefficient at x = ¡3 ; also no upper or lower left-hand differential 
coefficient at x= a. This fact must be borne in mind. We call 
the functions 3) dérivâtes to distinguisli them from the deriva 
tives Rf, Lf. When Rf (a) = Rf'(a), finite or infinite, 
RfW exists also finite or infinite, and has the same value. A 
similar remark applies to the left-hand differential coefficient. 
To avoid such repetition as just made, it is convenient to in 
troduce the terms upper and lower unilateral differential coeffi 
cients and derivatives, which may be denoted by 
Üf , Uf. (4 
The symbol U should of course refer to the same side, if it is 
used more than once in an investigation. 
When no ambiguity can arise, we may abbreviate the symbols 
3), 4) thus: 
R , R , L , L , Ü , U. 
The value of one of these dérivâtes as R at a point x = a may 
similarly be denoted by 
12(a). 
The difference quotient 
f(a) —/(&) 
a — b 
may be denoted by 
A (a, 5).