Given:
$f(x) + f(x+T) = 2$ ; where $T$ is a fixed positive number.
The solution is given as:
put $x = x+T$
then given equation becomes
$f(x+T) + f(x+2T) = 2$
subtract given equation from above. You'll get: $f(x) = f(x+2T)$.
Hence $2T$ is the period of $f(x)$.
I don't get it. wouldn't putting $x = x+T$ change the value of the function? How come we are still equating it to $2$? If the function value doesn't change then we are implicitly assuming that $T$ is the period right?