What is the meaning of a single value in a continuous density function.

By theory, probability of a single value must be zero. However, you
can give a value to standard normal density function for example, and
get a number from R: dnorm(0,0,1,FALSE) gives 0.3989423
This is the probability at point 0 for standard normal distribution.
The question is, although the function is capable of giving this
number, from a theory point of view, this number should be zero.
Would anyone care to comment on the meaning of this value? How do we
explain this contradiction? We have a value given by a function, but
it should be zero according to general rules of probability.

All the best
Seref
.

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