Is NaN falsy? Why NaN === false returns false

1.Why NaN === false => false, isn't NaN falsy?

The term "falsy" isn't defined in ECMA-262, it's jargon for where type conversion coerces a value to false. e.g.

var x = NaN;

if (!x) {
  console.log('x is "falsy"');
}

The strict equality operator uses the Strict Equality Comparison Algorithm which checks that the arguments are of the same Type, and NaN is Type number, while false is Type boolean, so they evaluated as not equal based on Type, there is no comparison of value.

2.Why NaN === NaN => false, but !!NaN === !!NaN => true

Because the strict equality comparison algorithm states that NaN !== NaN, hence the isNaN method.

Using ! coerces the argument to boolean using the abstract ToBoolean method, where !NaN converts to true and !!NaN converts to false, so:

!!NaN === !!NaN  -->  false === false  -->  true

Note that the abstract equality operator == will coerce the arguments to be of the same Type according to the rules for the Abstract Equality Comparison Algorithm. In this case, NaN is Type number, so false is converted to a number using toNumber which returns 0. And 0 is not equal to NaN so:

NaN == false  -->  NaN == 0  -->  false

This condition:

NaN === false

Is always false because numbers are not booleans. To test if a value is falsy you can use a ternary expression:

NaN ? "truthy" : "falsy" // falsy

Why NaN === NaN => false

This is explained in MDN; pragmatically speaking, though, two values of which you only know they're not numbers can't logically be the same thing.

... but why is !!NaN === !!NaN => true

This is because casting NaN into a boolean will make it false and booleans can be compared as per normal.

1. Falsy and being strictly equal to false are very different things, that's why one has a y instead of an e. ;)
2. NaN is spec'd to never be equal to anything. The second part of your question is comparing false === false, which is funnily enough, true :)

If you really want to know if something is NaN, you can use Object.is(). Running Object.is(NaN, NaN) returns true.