Consider a morphism of schemes $f \colon X \to Y$ having the following properties:

i) Topologically, $f$ is a homeomorphism onto its image.

ii) The induced map of sheaves ~~$\mathcal{O}_Y \to f_* \mathcal{O}_X$~~ $f^{-1}\mathcal{O}_Y \to \mathcal{O}_X$ is surjective. (See Wedhorn's answer.)

iii) $f$ is [edit: locally] of finite type. (This excludes morphisms such as $\mathrm{Spec} \; K(X) \to X$ with image the generic point of the variety $X$.)

Clearly, any immersion satisfies these three properties.

I have two questions:

1) Is $f$ necessarily an immersion (i.e., closed immersion followed by an open immersion)?

2) If not, are there important properties satisfied by immersions, but not necessarily by morphisms satisfying the three properties above?

Motivation: I find the three properties above more "natural" to work with than the property of being a closed immersion followed by an open immersion. For instance, it is immediately obvious that the composition of two morphisms satisfying i)-iii) also satisfies i)-iii); I do not find this immediately obvious for immersions in the usual sense (although I will be pleased, if not entirely surprised, if some respondents make this obvious).