Proof
Each of these follows by utilizing the corresponding linearity properties of the summation and integration operations. For example, to verify part three in the continuous case:
\begin{align*}
E[v(X) + w(X)] & = \int_{x \in R} [v(x)+w(x)]f(x) dx\\
& = \int_{x \in R} v(x)f(x) dx + \int_{x \in R} w(x)f(x) dx\\
& = E[v(X)] + E[w(X)].
\end{align*}