Necessary condition for the integral equation $u(x) = f(x)+\lambda\int K(x,t)u(t)dt$ to have a continuous solution? Is the following condition necessary for the integral equation $$u(x) = f(x)+\lambda\int K(x,t)u(t)dt$$  to have a continuous solution: $f(x) \neq 0$, is real and continuous in the interval $[a,b]$?

When $f(x) = 0$, the integral equation will become homogeneous. So I think this should be a necessary condition.  Am I right? Please suggest me
Why would $f$ have to be nonzero or real? If $f = 0$ there is certainly a continuous solution, namely $0$.  There may be nonzero solutions as well.  You'll probably want to assume $K(x, \cdot)$ to be in $L^1(a,b)$ so the
integral exists for all $u \in C[a,b]$.  Then under some conditions on $K$ (e.g. if $K$ is continuous) $\int_a^b K(x,t) u(t)\ dt$ will be continuous for all continuous $u$.
If so, $f(x) = u(x) - \lambda \int_a^b K(x,t) u(t)\ dt$ will have to be continuous in order for there to be a continuous solution, and will have to be real if $K$ is real and there is a real solution.
It has a unique continuous solution.It's solution is given by Lioville-Neumann Series.
          http://en.wikipedia.org/wiki/Liouville-Neumann_series