A likelihood function equal to a Dirac delta is a degenerate case, as the Dirac delta is not a function. It would mean that, after the observations, you are completely sure about the value of the parameter and all possible uncertainty is removed. In theory you can deal with such cases, but in practice, they are imposible. If your question arise from a practical situation, the observation model should be reviewed.

For such likelihood functions, discrepancy analysis is also degenerate, as the final (posterior) distribution of the parameter/s is concentrated in a single value of the parameter/s.

I would echo the comments of Professor Egozcue. I will add that the general issue you raise may relate to nonparametric Bayesian models. 

A couple of potentially useful links...

http://www.stat.columbia.edu/~porbanz/npb-tutorial.html

( a sweeping view that goes from current research to historical origins of Bayesian nonparametrics)

http://www.stat.tamu.edu/~sk/teaching/GhosalDP.pdf

(a more focussed, but non-trivial mathematical treatment)

http://people.ee.duke.edu/~lcarin/nDP.pdf

(one paper with an illustrated application)

If X and Y are continuous random variables, and Y is a deterministic function of X, e.g., Y = g(X), then the conditional PDF of Y given x can be written as DiracDelta(y - g(x)). Thus, if f(x) is the PDF of X, then f(x) DiracDelta(y - g(x)) can be regarded as the joint PDF of (X, Y), from which you can compute the marginal PDF of Y by integrating X out of the joint.

Thank you for your valuable responses.