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- Homework Statement
- Prove that

##f(x) = \left \{ \begin{matrix} x^2 & \mbox{if }\;x\in \mathbb Q \\ 6(x-3)+9 & \mbox{if }\;x\in \mathbb R\setminus \mathbb Q\end{matrix}\right.##

has a derivative only at ##x=3##.

- Relevant Equations
- Analysis, algebra, number sets

(a) ##f(x)## is continuous only at ##x=3##:

1- If ##x\in\mathbb Q##, ##f(x)=9## at ##x=3##; around, there is ##\mathbb Q##

2- If ##x\in \mathbb R\setminus \mathbb Q##, this is the set of irrational numbers.

Intuitively, if ##x## was in ##\mathbb R##, ##x^2## and ##6(x-3)+9## would meet at ##x=3##; but, around ##x=3##, there are ##\mathbb Q## and ##\mathbb R\setminus Q##.

(b) If ##f:\mathbb R \to \mathbb R##, it's easy to prove that ##f(x)## has a derivative at 3: 6.

Greetings!

1- If ##x\in\mathbb Q##, ##f(x)=9## at ##x=3##; around, there is ##\mathbb Q##

2- If ##x\in \mathbb R\setminus \mathbb Q##, this is the set of irrational numbers.

Intuitively, if ##x## was in ##\mathbb R##, ##x^2## and ##6(x-3)+9## would meet at ##x=3##; but, around ##x=3##, there are ##\mathbb Q## and ##\mathbb R\setminus Q##.

(b) If ##f:\mathbb R \to \mathbb R##, it's easy to prove that ##f(x)## has a derivative at 3: 6.

Greetings!