Why not say the buyer will *definitely* buy at price X? In supply-and-demand curve, we draw curves of indifference. Say the demand goes through (1, \$1). Then Bob the Buyer is indifferent between spending \$1 for 1 apple or just keeping the \$1. If Sally the Seller had 1 apple to sell, then the equilibrium quantity sold would be 0 or 1—we cannot say. Seen differently, Bob will buy 1 apple if the price is in the interval [\$0, \$1) (and he may buy it if the price is exactly $1).

Would it not be simpler to use closed intervals? In our example, Bob will buy 1 apple is the price is in the interval [\$0, \$1]. The equilibrium quantity sold will definitely be \$0. Seen differently, the demand curve represents the maximum price buyers will pay, and the supply curve represents the minimum sellers will sell for.

It's (literally) an infitesmal of a difference, but it seems, at least for discrete situations like the above, it would be nice to say exactly what will happen.

Why don't we do that? There must be some elegant reason.
Under Yatharth's approach, the set of prices at which Bob will definitely buy an apple is the closed interval $[\$0,\$1]$, which is nice. However, the set of prices at which Bob will definitely not buy an apple now becomes an open interval $(\$1,\infty]$.

In contrast, under the conventional approach, the set of prices at which Bob may buy an apple is closed: $[\$0,\$1]$. So too is the set of prices at which he may not buy an apple: $[\$1,\infty)$. So, under the conventional approach, we're dealing solely with closed intervals.

Both approaches have their merit. But I believe that all things considered, fewer inconveniences arise with the conventional approach than with your suggested approach. In general, it's "nicer" to not have to deal with any open intervals.

Example 1. Under Yatharth's approach, there is no answer to the question "What is the lowest price at which Bob may not buy an apple?"

Example 2. Under Yatharth's approach, the demand curve will be discontinuous and with jumps. In contrast, under the conventional approach, it is continuous (albeit with kinks).



Elaborating on Example 2.

The demand function $D:\mathbb{R}_{0}^{+}\rightarrow\mathcal{P}\left(\mathbb{R}_{0}^{+}\right)$ maps each price (a non-negative number) to a set of quantities demanded (a subset of the non-negative numbers). The economic interpretation is that at each price, $D$ tells us what numbers of units of the good the consumer most prefers to buy.

We assume that the good is infinitely divisible.

Then here are the graphs of the demand function under the two approaches:



The demand function is continuous under the conventional approach, but not under Yatharth's. (Nonetheless, under Yatharth's, it is lower semi-continuous.)