transfer function pols
If your stimulus  is a sinewave at the corner frequency, you will see an output sinewave that's 3db lower than the input.

The transfer function doesn't "blow up" because you're not causing the denominator to become zero.  You're looking at the special case where s=jw, (i.e. a pure sine on the jw axis) and w is the corner frequency. 

Plug that (s=jw) into the transfer function and take the magnitude and you'll see that the output is just a sinewave with the 3dB attenuation.