How to differentiate integral functions .

Let E c R"

be a measurable set.

Let A C Rh be an

open set and let f : Ex A -0112

( × , t ) |-O f- ( × ,t) -

Ft e A define Fct ) : = ffcxit) dx

E

Lemme : Let f be L - measurable in x tte A

and continuous in t for a . e. ×EE.

Assume that

Fge L' (E) such that If Cx ,t)|< g ( x ) tt EA,

a. e. XEE

Then F is continuous in A.

Proof x to EA,

Ltn } CA .

.

tn -0 to as n → + x.

since fed in t we have fcxitn ) → fcxito ) a. e. xeE,

Moreover lfcx.tn ) | E gcx ). By dominated convergence

Fctn ) → Flto ) D

Theorem : Let fe L' (E) t tea and fed CA ) a. e. xet.

Suppose Zgn , g. ,

-

, gy⇒ e t (E) :

t.fr#jknt)/=gjk), je { 1,2 , - in } ttea

,a. e. xee

Then Fed (A) and II. Ct ) = § {¥jC×Hok .

Proof . We prove the theorem for k=1 ( ACR ).

Let

t.CA,

r > o : 13 ( to ,r ) CA, ltn }ncBC to ,r ) : the → to

.

Fetal - flto )= § fkftytft#took = § Ft ( ×

' %) dx

the - to

for some 5€ B ( to,

r ) ,since of e C

'

(A) . By the continuity

of -2€we have also that 2¥ Cx , Ee ) → 0¥ G. to ) as

k → tx,

a. e. xe E. Moreover I ¥f ( x

, E) | e g Cx ) et (E)

By the dominatedconvergence theorem we have that

lim fctu ) - Fcto )k→+ot=t = fin § ¥tG' %) 0k¥ ¥+4 ' took .

We have that F'

Ct ) = § 3¥ ( x. t ) ok. By the Lemme with

2¥ in place of f we have that F'

is continuous in A,

that is Fe C'

CA ), ]

In the case K > I the proof is the same : just consider

derivatives in each of the k directions and argue as above-

FmF : A x [ a , b) × [ aib ] -0112 defined as

pFCT , d

, p ) : = f fcxit ) ok,

with F eCh

.

&

Z±y.

C tide ) = §2¥j⇐Hd× ; TIP = ftp.t ), IT = -f÷

fundamental than of calculus

With a ,p : A → Caib ] : t.CA t > the [ anb ] are

tea - > p HE [ a ,b]

C'

(A) functions,

we now consider Gct ) : = Flt ,ah

, PHD .

putWe have GK ) = / fk , ⇒ ok .

& Ct )

By the previous fhm we know that G is the composition

of C'

- functions in tent - variable.

Hence we

may write

Fg .

Ct ) = }zjF ( tidal, put ) =

= Ft ( tidal, pH ) ) + Fattixkl , put )§§CHtFp(

THAI

,pttD2,¥!t)

; .

pct )

= / 2¥,

( x. ⇒ ok -

fcacttttofdzjattfcpcttt) }¥Ct).

Lct )

Dnthe simpler case A CR

,one can write

It §dYtcxH°k=faY¥¥⇐Hok+f(pHHp' at - fatty Ita.