Dear Rajesh, first of all, different questions have different answers. So the answer below is purely for the actual question about the pressure that you stated and may be wrong for many other, "similar" questions that deal with other functions. You're very naive if you think that all questions about functions in physics have the same answer.
Pressure is a macroscopic concept because gases etc. are made out of atoms. So the number of atoms or molecules in a volume of space is an integer. If the volume is really small - if you really want to define the pressure at an "accurate locus" of space - then the integer-valuedness becomes really important and the pressure is behaving as a discrete, discontinuous variable.
However, if one averages the pressure over a large enough region of space so that the number of atoms in this region is much greater than one, then the discontinuities in the number of molecules - and their velocity - are suppressed by the statistically large number of the molecules, and the assumption that the pressure is approximately a continuous function of time becomes an acceptable approximation.
Classical field theory is a good model of the reality - at least some portions of it - and it is based on continuous (and differentiable) functions of space and time.
The discreteness above - that arises from the atomic structure of matter - isn't true for other things. If you ask what is the electric field at a given point, it is classically a totally smooth function and there is no disclaimed at all. However, quantum mechanically, the actual values of the quantities don't exist prior to the measurement. There are lots of subtleties.
Various layers of physics are more or less accurate and neglect various things. Correspondingly, some objects may be continuous, discontinuous, differentiable, or non-differentiable in different descriptions. And in quantum mechanics, there doesn't really exist any objective function that describes the state of the system at each moment at all. Even though the electron is point-like, at least with the accuracy of $10^{-18}$ meters, one can't say what was its trajectory through space, $x(t),y(t),z(t)$, at the same accuracy.
Best Answer
An important example in quantum mechanics is e.g. the Hilbert space
$$H~=~L^2(\mathbb{R}^3)$$
of Lebesgue square integrable wave functions $\psi$ in the position space $\mathbb{R}^3$. The Lebesgue square integrable functions (as opposed to just the Riemann square integrable functions) are needed to complete the Hilbert space with respect to the square norm
$$||\psi||_2~:=~\sqrt{\int d^3x ~ |\psi(x)|^2}.$$
Concerning completeness, see also this Phys.SE post.