1. Functional equations in one variable
1. Functional equations in one variable
1. Functional equations in one variable
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Mathematical Database<br />
Now, as f is cont<strong>in</strong>uous, we can always bound an irrational number by two rational numbers. (For<br />
example, we may bound π by 3, 3.1, 3.14, 3.141 and 4, 3.2, 3.15, 3.142 respectively). Let<br />
{ xn},<br />
xn∈ be such a sequence with lim x n<br />
= x , then by cont<strong>in</strong>uity of f, we have<br />
n→∞<br />
f ( x) = f(lim x ) = lim f( x ) = lim cx = cx. Therefore f ( x)<br />
n n n<br />
n→∞ n→∞ n→∞<br />
= cx for all x∈R .<br />
Q.E.D.<br />
Corollary 3.2.<br />
If f is a cont<strong>in</strong>uous function and for all xy∈ , ,<br />
(i) f ( x+ y) = f( x) f( y)<br />
then f ( x)<br />
= c<br />
(ii) f ( xy) = f ( x) + f ( y)<br />
then f ( x) = clnx<br />
(iii) f ( xy) = f ( x) f ( y)<br />
then f ( x)<br />
= x<br />
where c is a constant.<br />
x<br />
c<br />
Example 3.3.<br />
If<br />
f :(1, +∞)<br />
→ is a cont<strong>in</strong>uous function such that f ( xy) = xf ( y) + yf ( x)<br />
for all1 < xy , ∈ , f<strong>in</strong>d<br />
f ( x ).<br />
Solution.<br />
It is equivalent to f ( xy) = f ( x) + f ( y)<br />
. If we let<br />
xy x y<br />
gxy ( ) = gx ( ) + gy ( )<br />
f ( x)<br />
gx ( ) = , then the equation becomes<br />
x<br />
which is simply (ii) of corollary <strong>1.</strong>32 (although there is a m<strong>in</strong>or difference<br />
between them). So we have g( x) = clnxand<br />
f ( x) = xg( x) = cxlnx.<br />
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