74 Transactions of the Academy of Science of St. Louis 
f(x) = =, Qtrin}(X — Xo)” = a Dirzn}(% — Xo)”, 
0 0 
then @[r4nj =5,r4n) for all n. 
MOes AE TCR) =) ,4 [r¢n](X—2X9)” throughout the sphere of 
convergence of the right-hand member, then the derivative f’ (x) 
is also regular in the same region and f’(«) =) na tr+nj(%—Xo)”. 
1.04. TE fie) = >a [r+n](X —Xo)” throughout the sphere of 
convergence of the right hand member, and [,, is a uniform arc 
entirely within this sphere, then 
b x b 
f fix)dx = Ba @tr4nj(% — Xo) "dx 
Ta 0 Ta 
Gtrn){(b — xo)"*2 — (a — xo)*{ 
n+1 
or, 
f Kar = F® — FC, 
where f(x) = F’(x). 
In T.33 it was seen that a function which is regular is differ- 
entiable. Does the converse hold? The answer, which is in 
the affirmative, may be found from a theorem established by 
Gateaux.* He has shown that a function which is differentiable 
throughout the p neighborhood of a point may be expressed by 
an infinite series), U,,(x —xo) which converges uniformly in the 
given neighborhood, and is such that U,,(x—<x) is a function of 
the form discussed in the Note (see p. 37), and called by Gateaux 
a homogeneous polynomial of order m. But we have seen that 
such a function, on En) to E;,) for instance, may be expressed 
in the form @1,4n}(x—xo)"; therefore, the given function may be 
represented by a power series 5 aa (*% —x9)",which converges 
uniformly for ||x— | < <p. On combining this result with T.33 
we have 
T.35. A necessary and sufficient condition that f(x) be regu- 
lar in a neighborhood of a given point is that it be differentiable 
in the neighborhood. 
Moreover, Gateaux’s result proves a sort of a converse to 
T.21, for it shows that if f(x) is regular at every point inside a 
* Sur diverses questions du calcul fonctionelle, Soc. math. de France Bull., 
p. 21, vol. L, 1922. 
