Find $a$ and $b$ using the precise definition of limit

Question

Let $f : \mathbb R \to \mathbb R$ be a function defined by

$$f(x) = \begin{cases} x^2+ax+1 &\text{if } x \in \mathbb Q\\ ax^2+2x+b &\text{if } x \in \mathbb R \setminus \mathbb Q\end{cases}$$

Knowing that $f$ is continuous at $1$ and $2$, find $a$ and $b$ using the precise definition of limit.

I have no idea how to start.

Answer 1

Let's say $\alpha(x)=x^2+ax+1$ and $\beta(x)=ax^2+2x+b$.
So $f(x)=\begin{cases}\alpha(x)&x\in\mathbb Q\\\beta(x)&x\in\mathbb R\setminus\mathbb Q.\end{cases}$

This function can only be continuous in $x=1$ and in $x=2$ if $\alpha(1)=\beta(1)$ and $\alpha(2)=\beta(2)$. The rest I leave to you.
Convince yourself of the above. It is the key argument.