This is a comment but will be too long for that. The equation given is a (very simple) example of a difference-differential equation (in fact, a difference equation but there are several related queries on MO where derivatives are also involved--see, e.g., "Are there any nonlinear solutions to $f(x+1)-f(x)=f´(x)$ and "On equation $f(z+1)-f(z)=f´(z)$). Now there exists, not unexpectedly, a comprehensive theory of such equations, in particular, explicit solutions to the case of linear equations with constant coefficients, of which the above equations are very special cases. One can, for example, consult the comprehensive monograph "Differential-difference Equations" by Bellman and Cooke, readily available online.
This situation is quite common on MO, not surprisingly since the most prolific active participant tend to be problem-solvers, who attack such problems sometimes without knowledge of current, relevant theory. This is, of course, not a bad thing. However, it seems to me to be a truism that if such a
general theory is known by a contributor to exist, then it is a matter of common courtesy and professional integrity that this be acknowledged in the following discussion. Hence this comment.
One field which is fraught with this phenomenon is that of (Schwartzian) distributions and this is relevant to this example since a standard and simple way of obtaining all solutions to equations such as the one mentioned here is the use of the Fourier transform. One frequent MO canard is that one can only take the FT of tempered distributions. In fact, all distributions have such transforms, it is just that they can be of more general forms, namely analytic functionals, i.e. continuous linear functionals on suitable lc spaces of entire functions, in particular, delta distributions with singularities in the complex plane. This is particularly relevant to the above simple concrete examples, a fact which will not surprise anybody who is familiar with the methods of solving second order linear ode´s with constant coefficients (basically every professional mathematician).
As a final comment, all of the three cases mentioned here can be answered in seconds using FT´s, sometimes obtaining distribution solutions which remain undetected by other methods. This happens in the case of the difference equation of this query, which exhibits solutions not mentioned in the answers. These are, of course, precluded by the stipulation that the solution be a rather special kind of distribution, a continuous function on the line, but they are perhaps worth mentioning since they might be useful in another context.
