Q:

Diode Connected MOS

let's consider the equivalent circuit of a diode connected Mosfet.
Since the current source between drain and source is:
gm * Vgs = gm * Vds
and it is proportional to Vds, it means that this current source behaves like a resistor of value 1/gm.
Now I have two questions:
- a diode connected Mosfet works in saturation, that means as a controlled source. How is it possible that it behave like a resistor (which is what will happen in triode region)? 
- Mosfets used as amplifiers need to be polarized in saturation in order to generate a current which is controlled by a voltage signal. So may be a diode connected Mosfet useful for an amplifier? Or is it just like a resistor and so unuseful for that aim?

A:

How is it possible that it behave like a resistor (which is what will
  happen in triode region)?

You can’t equate it to the triode region unless you are holding Vgs at some fixed value and, you can’t do that because Vgs = Vds when wired as a MOSFET diode.

may be a diode connected Mosfet useful for an amplifier?

If it’s connected as a diode there are only two terminals and this means, I suspect that it is impossible for it to be an amplifier (else we’d be using regular diodes like this all the time).

