# Math Help - well defined

1. ## well defined

Let $f,g:R \rightarrow R$ be Lebesgue integrable functions. Let $\phi(x,y)=f(x-y)g(y)$. Show that $\phi$ is well defined.

i am not sure what it means by a function is well defined. would someone tell me what i have to show to say that it is well defined? any help would be appreciated.

2. Well-defined means that for each input of a function, there is one output. For example, a circle is not a well-defined function.

First, note that since $f,g$ are Lebesgue integrable, they are well defined.

Also, $x=x',y=y'$ if and only if $(x,y)=(x',y')$.

Then, we have that since $f,g$ are well-defined, then $f(x-y)=f(x'-y')$, on account of $x-y=x'-y'$. Similarly, $g(y)=g(y')$.

Hence $\phi (x,y)=f(x-y)g(y)=f(x'-y')g(y')=\phi (x',y')$, and $\phi$ is well-defined.