Undergrad Discrete to continuum Gaussian function

[Aleolomorfo]

I have a question regarding a paragraph in "Radiation detection and measurement" by Knoll.
In the chapter about the discrete Gaussian it states that "Because the mean value of the distribution $\bar{x}$ is large , values of $P(x)$ for adjacent values of x are not greatly different from each other. In other words, the distribution is slowly varying". Then it states that, because of this property, we can modify the discrete Gaussian to a continuos Gaussian.
I do not understand the link between the two statements.

 

[BvU]

Don't know about the context, but it seems to me this refers to the analysis definition of continuity (*), to justify using a function to describe a discrete distribution. Something like "if we let the number of observations go to infinity, the relative difference between the discrete and the continuous description will go to zero"

(*) For all $\varepsilon > 0$ there is a $\delta > 0$ such that ... etc.

 

[Klystron]

@BvU
Found this pdf file with a similar textbook https://phyusdb.files.wordpress.com/2013/03/radiationdetectionandmeasurementbyknoll.pdf

Written by Glenn Knoll. Gaussian distributions examined circa page 76.

 

[BvU]

Yes. Indeed, he goes from binomial via Poisson to Gauss, initially only 'defined' for discrete $x$. Then generalizes to a continuous Gaussian. That wouldn't work if the discrete function would not smooth out (e.g. as with the function int(x) )

 

[WWGD]

But aren't PDFs required to just be piecewise continuous? I mean, we do know the Gaussian is continuous, even smooth. Maybe the author is assuming this?