Why is the Empirical Distribution based on the Cumulative Distribution? Why is the Empirical Probability Distribution Function  based on the Cumulative Probability Distribution Function?
I have always seen the Empirical Distribution Function to have a "staircase" style shape, similar to a cumulative probability distribution function:

Is there any reason that the Empirical Distribution Function always appears in the shape of a "staircase" and estimates the Cumulative Probability Distribution Function instead of the Non-Cumulative Probability Distribution Function? Or is this a trivial matter - can the Empirical Probability Distribution be easily transformed into a Non-Cumulative Probability Distribution Function?
Can someone please explain this?
Thanks!
Note: Apparently F-hat (t) is an unbiased estimator of the true Cumulative Probability Distribution function F(t).
References:

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*https://en.wikipedia.org/wiki/Empirical_distribution_function

The empirical distribution function $\hat{F}(\cdot)$

*

*is a step function by construction, since it puts a probability (Dirac) mass of $1/n$ on every term in the sample, $(x_1,\ldots,x_n)$, hence jumps by the same factor $1/n$ from one observation to the next. As a result, it is not everywhere differentiable and cannot be associated with a probability density function, meaning there is no density equivalent to the empirical distribution function (at least wrt a continuous measure like the Lebesgue measure).


*is "empirical" in the sense that it is based on an iid sample, $(x_1,\ldots,x_n)$, as opposed to the true cumulative distribution function, $F(\cdot)$, of the sample


*is a proper cumulative distribution function (cdf), corresponding to an average of Dirac distributions over the sample, $(x_1,\ldots,x_n)$


*estimates the true cdf $F$ with the property of being unbiased that$$\mathbb{E}^F[\hat{F}(x)]=F(x)$$for all $x$'s, and


*is converging a.s. to the true cdf $F$ for the uniform convergence norm. This is the Glivenko-Cantelli Theorem.
What is a non-cumulative distribution function?
