real analysis - Can the function defined in this way be everywhere discontinuous?

Suppose that we have some real function of a real variable $f$ defined on the set $[a,b]$ which has the properties that:

1) $f$ takes values in the set on which it is defined

2) for every $y \in [a,b]$ there exists one and only one $x \in [a,b]$ such that $f(x)=y$.

The question is:

Can such function be everywhere discontinuous?

The question can be asked also in this form:

Does there exist everywhere discontinuous bijection defined on the set $[a,b]$ which also takes values in the set $[a,b]$.

I guess that the answer is yes but at the moment I am not smart enough to prove the existence or to construct such an example.

Thank you for your response and co-operation.