# Random Thought

Suddenly I want to post something about mathematics. It’s calculus, actually.

Evaluate the integral of a single dot: it doesn’t matter. It’s zero. Evaluate the integral of countably many dots: it still doesn’t matter. Still zero. How countable are we talking about? As much as rational number in real number, maybe. Now, take a continuous function, and evaluate the integral. Remove a single dot in it: doesn’t matter. The integral is the same. Keep removing the dots, countably many. The integral is still the same. Cool, isn’t it?

It seems like the integral doesn’t get affected by the adding and/or removal process, as long as it’s done countably many. Well, if you don’t get the idea of countably many, just think of natural number (which is 1, 2, 3, and so on). Until what terms did it end? Never. It’s just keep getting bigger and bigger. Fun fact is: natural number, integer, rational number have the same cardinality.

Just imagine, if you have a continuous, real-valued function, say $f(x) = 1$ in $[0,1]$, we get 1 as the integral. Now, you remove all of the rational number in it, you will get a curve but it’s uglier, because now your curve should be too ‘holey’. The integral is still 1.

In a cooler way to express why that happens, it’s due to Lebesgue Integrable Criterion.

Now I have plenty exam sheets awaiting to be graded, due to tomorrow. So I should be doing that by now. Well.. wish me a good luck! 😀