How to show the function is not Turing computable?
Since I assume this is a homework question, I won't give a fully detailed proof but rather a sketch with some gaps to fill.

Consider the function 
$$g(x)=
\begin{cases} 
  x+1 &amp; \text{ if } \varphi_x(x)\downarrow\\
  \uparrow &amp; \text{ otherwise}.\end{cases}$$
This function is computable (convince yourself of that!), so it has an index $e$. 

Now consider the function $f$ on $e$, then computing $f(e)$ comes down to verifying whether $\forall n\  \varphi_e(n)\uparrow$. Can you see why this contradicts that the halting problem is not computable?