For which values of x f(x) is derivative? What's the value of the derivative when exists?
How can I even begin solving such problem ?
Perhaps there is some confusion about what we're trying to do. From the way you worded the question, I believe we are supposed to identify what the derivative of is.
First, let's look at where is continuous. At these values, the derivative might exist. It cannot exist at values where is not continuous.
It's important to know a few things about irrational and rational numbers. First, between every two distinct irrational numbers, there exists a rational number that lies between them.
Also, between every two distinct rational numbers, there exists an irrational number that lies between them.
I'm not sure how best to explain this, but the implications of this is that is not continuous except where . Therefore the derivative can only be defined at that single point.
Next we verify that at .
These values are equal, so the derivative does exist at that point. Thus: