Gokyo Shumyo, Section 1, Problem 40 / Solution

Black to live  

is the vital point. It clearly secures two eyes at the circled points.

Continuation 1  

Note what a difference an extra white stone at a would make.

Continuation 2  

Again, an extra stone at a would make the difference.

Failure 1  

One is tempted to cover the two-space gap with a move like here. However, White strikes at the vital point after reducing the eyespace. at is met by at .

Failure 2  

The other ceiling defense at is defeated by and !. at a also kills with more difficult variations.