are fibres of a flat bijective map reduced? Let $f: X \to Y$ be a flat map of algebraic varieties or of complex analytic spaces which is bijective on closed points (or just bijective in the secnond case). Suppose  both $X$ and $Y$ are reduced. Is it true that $f$ has reduced fibres?
If it is true, I would be most grateful for a reference. 

If $D=Spec(\mathbb C[T]/T^2)$ is the double point and $P$ is the simple point, the morphism of analytic spaces $D\to P$ has reduced base, is flat and  bijective but has non reduced fiber.
