On the theorem of frullani

Theorem 4 (Pfeffer). If f: [a, b] —» R is Denjoy-Perron integrable on [a, Z>] being
F(x) = (DP) Г f(t)dt,
J a
and g is a function of bounded variation on [a, b], then the product f g is Denjoy-Perron integrable and
(DP) [^h f(t)g(t)dt = F(b)g(b) - F(a)g(a) - Л F(t) dg(t).
J a J a
We denote by (DP) f(t)dt the Denjoy-Perron integral of f on [a, b}.
We say that f: R -+ C is a locally Denjoy-Perron integrable function when, for every bounded interval [a, b] c R, f is Denjoy-Perron integrable on
[a, Z>]. If f: R —» C is locally Denjoy-Perron integrable, we define T_f\ 2 -+ C by
/^+o° . .
f(t)
-oo
Note that T_f is a distribution because if sup(9>) c (a, b), by Theorem 4, we can write
/+ОО fb rb
f(t)
-oo Ja Ja
and, since F is continuous (see R. M. McLeod [18] p. 58), we have


< A/_{e J},sup|/(t)|.
r^b
\T_f(
' J a
Now, if f is differentiable at every point, f¹ is locally Denjoy-Perron inte­grable (see R. M. McLeod [18], p. 27). Moreover, = DT_f, because if
e 2, we have by Theorem 4:
/+ОО _r+oo
/(OH') dt = - f(t)4> (t) dt - (DT_f,
-oo J —oo
Theorem 5. Let f: R —> C Denjoy-Perron integrable. Then f is a tempered distribution.
Proof. Put F(x) = (DP) f* f(t) dt for every x € R.
By Theorem 4, for every (p e S,
rb rb
(DP) f(t)
J a J a
Observe that, as F is bounded and continuous and q> e S, the Riemann- Stieltjes integral /2^ P(t)dq>(t) is defined, and
lim F(b)
lim F(a)
b—>+oo a—^ — oo
So f(t)
is Denjoy-Perron integrable on R and
/+ОО r + oo
f(t)
-oo J —oo
Now, since | F(t)
Msup,_eR |^'(r)|, we see that the linear form

I-» (DP) f(t)
is a tempered distribution.
As we want to extend the proof of Theorem 2 to the Denjoy-Perron integral instead of the Lebesgue integral, we need S. Lojasiewicz’s definition [ 17] of the value of a distribution at a point. It is said that a distribution T has a value at a point a when there exists the limit
lim T(a + kx) л^о
in the space 2' of distributions. This limit, when it exists, is a constant distri­bution, and we say that this constant is the value of T at a (see S. Lojasiewicz [17] or P. Antosik, J. Mikusinski and R. Sikorski [3]).