TITLE: Are there any nonlinear solutions to $f(x+1) - f(x) = f'(x)$?
QUESTION [12 upvotes]: Are there any nonlinear solutions to $f(x+1) - f(x) = f'(x)$?
(Asked by bcross at math.iuiui.edu on the Q&A board at JMM.)

REPLY [9 votes]: This is an elaboration of Qiaochu Yuan's prior comment: there are complex solutions (in fact, infinitely many) to $e^t-1 = t$, and then $e^{tx}$ is a solution.