Let $X=(X_1,\dots,X_k)$ be a random vector, such that $X_j\ne X_k,j\ne k$ and all components are non-degenerate and $a$ deterministic vector with non-zero coordinates. If $X_1$ is continuous random variable, does it follow that $a'X$ is continuous random variable?
What is nesessary and sufficient condition for $a'X$ be continuous random variable?
No; say $X_1,\dots,X_k$ are all independent, $X_1$ is continuous, $a=(0,1,\dots,1)$ and $X_2,\dots,X_k$ are all degenerate with the value $0$. Then $a'X$ is degenerate with the value $0$, and hence is not continuous.
IDK; there seem to be a lot of different cases, most of which are unrelated. For example, if $X$ is a continuous random vector then $a'X$ is a continuous random variable. But also, if $a$ has first component equal to $0$ and $X_2,\dots,X_k$ are all continuous, then $a'X$ is continuous.