I studied two important theorems on Borel measurable functions which are as follows:
$1.$ Every continuous function defined on a Borel set is Borel measurable.
$2.$ Monotone function defined on a Borel set is Borel measurable.
I want to understand why the assumption that the domain $X$ is Borel set important.For that I want to find counterexamples to the following statements:
$1.$ Let $X\subset \mathbb R$.Every continuous function $f:X\to \mathbb R$ is Borel measurable.
$2.$ Let $X\subset \mathbb R$.Every monotone function on $X$ is Borel measurable.
I think that to find counterexamples to this problem,I must find a non-Borel set.Can someone help me find the counterexamples because counterexamples always help to build intuition.