Example: a function accepts as input the source of another program (assume it's computable and halts). Thus, the original function is variable; can it be simulated by a UTM for any arbitrary valid input? I'm probably using the wrong terminology here, but the overall logic is what I'm getting at.
The input to a function can't change (it's a fixed set). If the input changes then you've defined a new function. That's the mathematical definition of a function. You're talking about a set/class of functions, so you would need 1 UTM per function.
