MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

3 added 501 characters in body

You probably want the function to be continuousely differentiable, and the convergence to be in the sense of the C1-norm (uniform convergence + uniform convergence of all the partial derivatives). You then also want the total derivative of F at the point x to be an invertible matrix.

Hmm...
Ah! Here's a way of getting to the same conclusion with much weaker assumptions:

If F is a continuous function and its derivative at the point x exists and is invertible, then it's enough to assume that the functions Fn are continuous and that they converge in the C0 norm (uniform convergence).
The reason is that F, when viewed as a map from a little sphere around x to the punctured space d-{0} has degree one. So any other map that is sufficiently close to it will also have degree one. A degree one map cannot extend to the disc bounding the sphere. So the function Fn must have a zero somewhere in that disc. Take smaller and smaller discs to finish the argument.