If the kernel is defined for x and y lying between 1 and 2, then the integral must go from 1 to 2 (not from 0 to 1), and the function space L must also consist of functions defined on [1,2], not [0.1].

If is an eigenvalue then , so . But as far as the integral is concerned, x is a constant, so we can take out the factor and get . But then the integral is a constant, and we are left with two possibilities. The first is that and the integral are both 0. The second is that f(x) is a nonzero constant times , in which case you can plug that formula for f into the integral, and you should then find that (if I've done the integration correctly).