Can a cpu still work if one single transistor breaks?

Depends on the design. Where the fault is. You'll remmeber one of
pentium computers had a hardware fault built in. That caused errors in
rare cases, but
they sold thousands before they pulled it.

The computers on the space shuttle 
were designed so one computer could be voted out by several other backup computers.

Some computers can work faster than designed, They are overclocked.

You mean the way CDs 'self repair' minor damage

- or RAM self corrects as it has a form of check bit, I can't remember
exactly, but intersecting triangles come to mind.

I doubt it in a CPU, those things are packed tight

As RedFox pointed out, for critical systems, you have 'voting'
- they are programmed in different houses
- the Airbus uses that 
- the only time I've been scared in an aircraft was when the pilot
pointed above his head and said 'there are three computers up there
and they vote'.

The Pentium hardware fault was a programming error.
- the hardware was fine, the logic was faulty - IIRC the math
coprocessor had faulty instructions for division.