If a is not in the image of f, and f is entire, then we know that there is a sequence in a domain whose value of function converges to a. Now the problem I faced is that even further we can choose a continuous curve p whose parameter is t in[0,1) such that p(t) goes to inf as t goes to 1, while f(p(t)) converges to a.
That's much stronger than image of entire function is dense in the plane .. Actually I faced two difficulties.
How can we find a sequence whose value of functions converges to a? With its modulus goes to infinity?
How can we connect these sequence to make a continuous curve whose value of function is not much varying between two points of consecutive points?