Proof that a function is not computable by a Turing Machine

I am going through a problem that I am asked to prove if there is a Turing machine that can compute f for every input a.

The function is the following :

$$f(a) = begin {cases} 1 & if there is some x such that M_a(x) halts with output 1 \
0 & otherwise end{cases}$$

I am struggling with the proof of this statement. I tried using a proof by contradiction and assumed there is such a Turing machine. I am also assuming I need somehow to use a universal Turing machine that takes $a$ as input but I am not sure how to proceed.

Can someone explain me if I am on the right track or any methodology I can use to prove or disprove this statement?