Couple of mistakes in the composition - black pawn where the dark square bishop is, dark square bishop on e7 instead of f6.
Is this position really resignable?

NMinSixMonths wrote:
krudsparov wrote:
?? White's winning, exf6 and white's a little ahead.
Even without his correction in the second post I think 1.fxe6 is far from winning for white.
On the contrary, I did state white was only a little ahead which is still winning but agreed, there's plenty of work to do to finish black off. With the corrections, I'd carry on, but more out of hope than expectation.

Maybe it is subjective, but I believe black is winning by quite a bit. Though black is a piece less, they have far more pawns than white, and all the pawns are connected.
Thanks for the feedback, and sorry again for the error in presenting.
@NMinSixMonths - I play around 1800 level, the student plays around 1600 level (up from under 1000 level less than a year ago), so it seems like the decision to resign was a reasonable one, in that context.
I tend to discourage premature resignation in club play, simply because playing out losing positions builds skills which can't be improved if you aren't under pressure. But I also know from experience that it isn't very fun to be slowly ground down and picked apart in a position like this by a player who knows what he's doing.
Once again, thanks to all!
White to play:
I was playing black, and white resigned. The player on white is nearly equal in skill to myself. Clearly white is under serious positional pressure, and is down 2 pawns. We analyzed this position in the club room, and it seem very bad for white, but I'm still wondering if there was some more precise defensive move (or counter) which could make it work out OK. Any thoughts?