Class of Total function is PRC Class ?

HI,

I am not sure if this is the right proof. If not can anyone please
point me to right direction.

Q. show that the class of all total function is a PRC class.

My Proof: We know that Class of Computable function is a PRC Class (by
showing Initial functions are computable). And All Computable
functions are Total and Partially Computable.

Hence Class of Total Function is a PRC class.
--------
But some seems odd here. coz all Total Function is not Computable
right ?

thanks


.