A discovered checkmate that blocks the check seems like something doable.
Getting out of check with checkmate
Here's a quick example I just made up:
(see http://www.chess.com/emboard?id=2318166 if the diagram is not showing)
If White plays Rxe5, the Black may see a mate in two with Rd1+, Re1, Rxe1#
But unfortunately for him, it is actually Re1#, mating the black king while blocking check on the white king

Nice! Though I think you mean Rxf6 to begin with. Also Rf1+ wouldn't be mate in 1 because of e5, Bxe5# but in principle it works. Thanks for working that out. That's been really bugging me!

A good example as it's an example where by delivering a check black loses on the spot. Kg7, h6 or h5 are the only moves that don't lose for black after Rxf6.

See, I told you so that a king can legally check and checkmate another king...
I always see the following:
Kc4+ or Kd2#

Not sure how that relates to this topic Jion_Wansu? Also a king can't technically check or checmate. A king move can lead to another piece delivering check by dicovered attack but the king itself is not actually the piece that checks.
Nice! Though I think you mean Rxf6 to begin with. Also Rf1+ wouldn't be mate in 1 because of e5, Bxe5# but in principle it works. Thanks for working that out. That's been really bugging me!
I tweaked the diagram after the original post because the first one had a way to block the check with a pawn move, which defeated the point. That's why I changed the text of the post as well.

Yeah the diagram makes sense with your moves now. I got the original diagram when I first looked for some reason.

Not sure how that relates to this topic Jion_Wansu? Also a king can't technically check or checmate. A king move can lead to another piece delivering check by dicovered attack but the king itself is not actually the piece that checks.
I know but technically the king declares check or checkmate on the opposing king legally, because the king made the last move

A retro problem to show the theme. No stipulation needed. That the position and solution is legal can be deduced solely from the board situation.
(Problem without words)
It seems rather obvious that the only way that could occur is to capture the piece offering check. Otherwise checkmate could not occur with a block of the check as the blocking piece could be captured by the piece offering check.

It seems rather obvious that the only way that could occur is to capture the piece offering check. Otherwise checkmate could not occur with a block of the check as the blocking piece could be captured by the piece offering check.
I beg to differ. Equally obvious that blocking can deliver checkmate if the checking piece was pinned.
Also simply moving the king may work.
I wonder if I'll ever get a chance to checkmate someone with a discovered attack from a piece that is pinned to my king and therefore technically unable to deliver the killing blow.
Example: http://www.chess.com/emboard?id=2319402
Black move Be4, forking the white queen and rook, but that leads to e3#. A discovered mate given by a bishop that is pinned, and the rook that is pinning the bishop is also pinned and unable to capture.
Is it technically possible to simultaneously get out of check and checkmate your opponent in one move? I feel like this should be possible but can't imagine a position where this might occur. Anyone know? If so do you have examples?