Is there a total recursive function $f:N \to N$ such that for no $\Sigma\_1$ formula $\phi(x,y)$ which defines it (i.e., defines its graph), is it true that PA proves that "$\phi$ defines a total function".
  

For instance, is this true for Paris-Harrington's function? (There, we know that if we take $\phi$ to define the function in the intended form, it cannot be proved to be complete, but maybe another formula will do the trick.)



PA proves that functions are total
There indeed are such functions. I cannot confirm that function arising from Paris-Harrington theorem has this property (though I suspect it does) but existence and other example follow from the following theorem:



> 
> If PA proves that a given recursive function is total, then this function is primitively recursive in $H\_\alpha$ for some $\alpha<\varepsilon\_0$.
> 
> 
> 


In particular, $H\_{\varepsilon\_0}$ cannot be proven total, no matter what formula we choose to define it. 


Mentioned result is proven e.g. in "Classifying the provably total functions of PA" by Weiermann. 


