If $f(r,t)=g(r),\,\forall r,t$ would that make $f$ and $g$ constant functions?

Question

I know that if $f(r)=g(t),\,\forall r,t$ then $f(r)=g(t)=constant$, but

If $f(r,t)=g(r),\,\forall r,t$, where now $f$ depends on $t$ would that lead to the same conclusion i.e. $f(r,t)=g(r)=constant$?

Answer 1

Not necessarly. Your expression $f(r,t)=g(t)\ \forall r,t$ means that the function $f(r,t)$ does only depend on the variable $t$, that is, the values of $t$ you plot in $f(r,t)$ doesn't contribute to the values of $f(r,t)$. But that doesn't mean that $g(r)= constant\ \forall r$.