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I'd like to ascertain the continuity of a mapping with respect to a parameter. Consider the function

$$ g(u(x)+\gamma v(x)) = \left\{ \begin{array}{l} 1 & u(x)+\gamma v(x) > 0 \\ 0 & \text{else}. \end{array}\right. $$

Here, $u$ is a piecewise constant function over the interval $(a,b)$ and takes values $u(x) = 1$, $0$, or $-1$. The parameter $\gamma$ is a real positive number. Furthermore, $v$ is $\mathcal{C}^1(a,b)$ where $\dot{v}(x) = 0$ for a finite number of $x\in(a,b)$. In other words, $v$ cannot be constant for a continuum of $x$ and it may not chatter.

Suppose $h(\gamma)(x)=g(u(x)+\gamma v(x))$ where $h(\gamma)\in L^2(a,b)$. My question is: is $h(\gamma)$ continuous in $\gamma$? I'd also greatly appreciate relevant references.


Disclaimer: I'm new to the whole functional analysis world so if I'm loose with my words, I apologize.

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