3.2 • THE CAUCHY-RIEMANN EQUATIONS 105" (3-l,7). The parti~ -derivati-WS ~"' ai'i°d' ~ exist, so the Jn~·· ~ue t_l}eorem. for
·Ieal func~ions <;if\, twq variables implie$ tl~at a value x~ ~ts between xo ~d
x 0 + 6..x ~uch ~fiat we ~-·WX;ite'the,fi.ts~,te~~ in 15-racleets'on·the right·'side.
of Equation (3117) as · ; · ' · .· · " · ,. "'·. ~. ·:. v·.. ·, ~···~ , .•· .<-··;· • • .. • ~ , • • r , .; i :. ,. " , ••
u(xo +Ax, yo+ Ay)-u(X(), Yo+ Ay) i:;: :i£.,(x*;yo + Ay) A~.
.. ·.. ' ;:,; ' .. \ ';~ ;.,, '. ,,. ~ ... , ,. :·"'.. ..· ' ' ... .-·· ... .:,. ~ ~.
. fi:irtheimore, as 14., "ahd uy,arC "c9~tinuods at· {io, Yo)_, .thei:e ·exi:sts,'.1};IUaI),tity 7
e1 such: ~~hat · · - ' ,,... ·. ··. ·.. ·.. ·.