MattydaleltzhrodrdbtyIn get, if F : EHR "

→ Rm is diffble,

then f'

II) : R"→ Rm

is liner,4lb diffble

,

ht ( f'

#D'

t ) is jst f'#) gan .

So its land fund to the dat smelling the f"

K).

Bit ↳hr - order patnlduwlny do mehe sense .. .

Tf f. EER"

→ IR is diltblde,th the patldnwts Dif : R

"→ R are do pts ; it try one diftble

,then we st seed - oar

patldnwtus Dijf = Di Djf.

If all Dijf are anti nws an E,

thensy fe CYE ,

"2).

More gulf , if F : EER"

→ Rm,

thin FECYE Rm) it ah opt Fi e C 4 ER ).

Now,the old statttht mixed paths commute Bit quite tne : an he Dijf ¥ D; if it they n't antis :

E± : ht Had = { ×YI÷yI' lxyttiao )lxiyl= 10,0)

Then an dude tht f , D ,f,Dzf are all ants ezhe ,

& tht D. A & D.A exist evguhe & re at. exceptat 10,0)

,

lat D.zflo, o ) = I & Da Ho, 0) = -1.

the B a sntf Mean Value Thmfr mixed portals :

thm : Suppose f : EERZ → R,

D,f,Duf exist at ay pt . f E . Tf R E E B adod rectangle wl sides pulled to the and

.

axes wl opposite verbs (a. b)& ( ath

,6th) ,define

Dlf,R ) = Hath

,btkl - ttnth

,b) - Ha, 5th) + flats ) .

Thu F (x. HER st.

Dlf,R) = hk ( D . , f) (x , y ) .

*jiltE-is)

Proof: First of all, ht ulttfttibth

) - ftt , b) . Bythe nsl Mrt

,Z xela ,

ath ) st .

ulathl - ula) = h n' lx) = h( D ,flx

,

btk ) - D,Hx , b))

of case,ulath) - ulal = fl ath , bth) - Hath ,b) - Ha, bth) - flat ) ) = Dlf

,R )

,so we he

DH, R) =h( D. Hx ,btu ) - D

,flx ,b))

Nn,ht # =D

,Hx

,t )

.

Then the MH ⇒ F yctbisth) st.

vlbthl - ulb) = kv ' ( y) = k( Da , Hxiy)),

so we can carbine to get

BH, R) =h I D, flx ,HH - DAH, 4) = hlvlbth) - HH) = hk Datlxiy) #

thm : Suppnf : EERZ → R,D , f , Duf ,Dzf exit an allof E ,

& Dz,f is at .

at (a ,b) EE .Thu Dhf exists at ↳b) &

ID, 2 f)laid = (Dz , Hk , 5)

Cat : Tf FECYE

,R),then Dz, f =D , zf .

ie,mixed parties can mte whenthey are continues

.