(1), because f(x)=(ax+b)/( 1+x? ) This is a strange function, so

f(0)=b=0，

f( 1/2)=(a/2+b)/[ 1+( 1/2)？ ]=(2a+4b)/5=2/5，

From b=0, a= 1,

Then the analytic expression of the function f(x): f(x)=x/( 1+x? )。

(2) The domain of the function f(x) is: (-1, 1),

On (-1, 1), take x 1, x2,-1

f(x 1)-f(x2)= x 1/( 1+x 1？ )-x2/( 1+x2？ )=[(x 1-x2)( 1-x 1x 2)]/[( 1+x 1？ )( 1+x2？ )],

Because-1

So f (x 1)-f (x2) < 0,

According to the definition of monotonicity,-1

So the function f(x) is increasing function at (-1, 1).

(3)，f(t- 1)+f(t)& lt； 0,

f(t- 1)& lt； -f(t)=f(-t), (f(x) is odd function)

Because the function f(x) is a increasing function at (-1, 1), so

t- 1 & lt； -t and-1

T< is t < 1/2 and 0.

So 0