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1. Functional equations in one variable

If xf ( x ) < 1, then by (ii) we have condition 2. Thus, we must have Remark. 1 xf( x) xf ( x ) = **1.** This means Mathematical Database is a fixed po**in**t larger than **1.** This aga**in** contradicts with f( x) 1 = . x The concept of fixed po**in**t is **in**troduced **in** this question. If f ( x) of f. It is not so commonly used but it is also a useful technique. = x, then x is called a fixed po**in**t 3. Some famous functional **equations** In this part, we will **in**troduce some famous functional **equations**. We may quote them directly **in** competitions. Theorem 3.**1.** (Cauchy equation) If f is a cont**in**uous function such that f ( x+ y) = f( x) + f( y) for all xy∈ , , then f ( x) = cx where c is a constant. Proof. We may use the technique menti**one**d **in** example 2.**1.** First, put f ( x) y = 1 and let x be a positive **in**teger. Then f ( x+ 1) = f ( x) + c, where c = f(1) . Thus = cx for positive **in**tegers x. It is easy to verify that (0) 0 orig**in**al equation. For negative **in**tegers, we replace x by –x and get f = by putt**in**g x = y = 0 **in**to the f ( − x+ 1) = f( − x) + c. Thus f ( − x) =−cx for positive **in**tegers x. Therefore we conclude that f ( x) = cx for **in**tegers x. Let m x = , where m, n are **in**tegers. Then we have n ⎛m+ 1⎞ ⎛m⎞ ⎛1⎞ f ⎜ ⎟= f ⎜ ⎟+ f ⎜ ⎟ ⎝ n ⎠ ⎝ n ⎠ ⎝n⎠ , thus ⎛m ⎞ ⎛1 ⎞ f ⎜ ⎟= mf ⎜ ⎟ ⎝ n ⎠ ⎝n⎠ . However, 1 ⎛1⎞ c = f ( ⋅ n ) = nf ⎜ ⎟ n ⎝n⎠ ⎛1 ⎞ c f ⎜ ⎟ = ⎝n⎠ n and we have ⎛1 ⎞ c f ⎜ ⎟ = ⎝n⎠ n . This implies f ( x) = cx for rational numbers x. . So Page 8 of 10

Mathematical Database Now, as f is cont**in**uous, we can always bound an irrational number by two rational numbers. (For example, we may bound π by 3, 3.1, 3.14, 3.141 and 4, 3.2, 3.15, 3.142 respectively). Let { xn}, xn∈ be such a sequence with lim x n = x , then by cont**in**uity of f, we have n→∞ f ( x) = f(lim x ) = lim f( x ) = lim cx = cx. Therefore f ( x) n n n n→∞ n→∞ n→∞ = cx for all x∈R . Q.E.D. Corollary 3.2. If f is a cont**in**uous function and for all xy∈ , , (i) f ( x+ y) = f( x) f( y) then f ( x) = c (ii) f ( xy) = f ( x) + f ( y) then f ( x) = clnx (iii) f ( xy) = f ( x) f ( y) then f ( x) = x where c is a constant. x c Example 3.3. If f :(1, +∞) → is a cont**in**uous function such that f ( xy) = xf ( y) + yf ( x) for all1 < xy , ∈ , f**in**d f ( x ). Solution. It is equivalent to f ( xy) = f ( x) + f ( y) . If we let xy x y gxy ( ) = gx ( ) + gy ( ) f ( x) gx ( ) = , then the equation becomes x which is simply (ii) of corollary **1.**32 (although there is a m**in**or difference between them). So we have g( x) = clnxand f ( x) = xg( x) = cxlnx. Page 9 of 10

- Page 1 and 2: Mathematical Database FUNCTIONAL EQ
- Page 3 and 4: 2 2 2 x f x = + x + + x f x ( ) 3(4
- Page 5 and 6: Solution. Putting y = 1, then Mathe
- Page 7: If f ( x ) =−x, from (2.4) we hav