pdf fo a partially non-continuous (truncated) pdf?

Question

Suppose we have a continuous random variable X with a distribution thats support is the real numbers, e.g. a Gaussian distribution $X \sim {\mathcal {N}}(\mu ,\sigma ^{2}).$ Suppose further that this random variable is transformed with a function that maps all non-positive values to zero and all positive values to other positive values, e.g. $$\phi(x)= \begin{cases} 0 & \mbox{if}~x\leq0 \\ x*0.5 & \mbox{if}~x>0\end{cases}$$ In a sense $\phi$ truncates $X$ but it does not throw the truncated part away but outs it on a specific value. But what is the pdf $g$ of the transformed random variable $\phi(X)$? Obviously $g(0)$ should be $F_X(0) (= \ldots)$ and the distribution is a mixture between a Dirac distribution at 0 and a partial Gaussian distribution ${\mathcal {N}}(0.5*\mu ,0.25*\sigma ^{2}).$ for the positive numbers. So the pdf $g$ should have the form $$g(x)= \begin{cases} 0 & \mbox{if}~x<0 \\ F_X(0) & \mbox{if}~x=0 \\ \frac{1}{0.5*\sigma {\sqrt {2\pi}}} e^{-\frac {1}{2} \left({\frac {x-0.5*\mu }{0.5*\sigma}}\right)^2} & \mbox{if}~x>0\end{cases} $$ Questions.:

1. Is g(x) correct?
2. Can this be expressed in terms of Diracs $\delta$? E.g. $g(x) = F_X(0)*\delta(x) + (1-F_X(0)) * \ldots$? What to put instead of the $\ldots$ - a previously defined function h or is there an elegant way using the $\mathcal {N}$ notation from the gaussian part?
3. How to prove that $g$ is a pdf? Is the integral computation $$ \int g(x) ~dx = \int F_X(0)*\delta(x) ~dx + \int_0^{\infty} (1-F_X(0))* \frac{1}{0.5*\sigma {\sqrt {2\pi}}} e^{-\frac {1}{2} \left({\frac {x-0.5*\mu }{0.5*\sigma}}\right)^2} ~dx $$ correct and straightforward?
4. Has this kind of transformation a specific name (a transformation that truncates part of the distribution and maps the truncated part to a specific value)? It should occur rather regularly if one writes down the math for all kinds of risk analysis (where measures taken usually take away parts of the risk and map it to zero and only the extreme part of the distribution, i.e. some sort of tail, is transformed).
5. Any literature recommendation on this?

Thanks to all who answer!