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Suppose I have random variables $(X,Y) \in \mathbb{R}\times \mathbb{R}^{+}$ with parameters $\alpha, \beta \in [-B,B]\times [0,B]$ for $B >0$.

I am trying to prove that the function:

$$ \left(Y-\alpha X^{\beta}\right)^2 $$

is continuous and measurable.

For continuity, I am not sure if there is another method other than to resort to the $\delta-\epsilon$ approach. I am trying to prove that the function above is continuous with respect to the parameters and its parameter space.

For measurability, I am trying to see if there is a larger theorem to appeal to rather than to do it by definition.

Does anyone have any ideas? Is there a theorem I can use readily here? Thanks!

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    Fo measurability, it is well-known that the composition $g\circ f$ of measurable $f$ and continuous $g$ is itself measurable. For continuity, I'm not actually sure if what you want to prove is true without additional hypotheses. – 2017-02-09
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    the expression $(Y-\alpha X^\beta)^2$ on its own is not a function (except maybe from $\Omega$ to $\Bbb R$)! Are you asking if $$ \Bbb R \ni \alpha \mapsto (Y-\alpha X^\beta)^2, $$ is continuous? – 2017-02-09

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