It's not just the function that becomes undefined, it's the entire program. There's literally nothing that the compiler is supposed to do following undefined behavior.
i.e. Is it undefined behavior if there are code paths that might access an array out of bounds, but at runtime it actually never would ?
With this assumption you might even deduce that
int f(int x) {
if (x != 42) undefined_behavior;
return x;
} int f(int x) {
return 42;
}