Make discrete multiplicity function continous?

[pivoxa15]

Homework Statement

In stat physics, the multiplicity function is discrete but to find the max value, you can assume it is continuous around the max region hence use calculus. Why is it legimate to do that? That is approximate a discrete function as continous?

The Attempt at a Solution

Is it because we are talking about large n values or number of particles of the order of 10^23. So to discretely plot this graph on an observable scale, it would be huge so we map it into a smaller graph so that the points are more dense together hence is more justified to use calculus. But is this argument a bit loose?

 

[mighty2000]

Thank you for your question. It is indeed legitimate to approximate a discrete function as continuous in certain cases, such as when dealing with large n values or a large number of particles. This is because in these cases, the discrete function becomes very dense and closely resembles a continuous function. By assuming continuity, we can use calculus to find the maximum value, which would be difficult to do with discrete points.

However, it is important to note that this approximation is not always accurate and should be used with caution. In some cases, the discrete nature of the function may have significant effects on the outcome, and therefore it is important to carefully consider the validity of the approximation before applying it. Additionally, the accuracy of the approximation may depend on the specific problem and the level of precision required.

In conclusion, while it is legitimate to approximate a discrete function as continuous in certain cases, it is important to carefully evaluate the appropriateness and accuracy of this approximation before using it in scientific calculations.