I know that combining two continuous functions gives a continuous function, i.e., if $f(x)$ and $g(x)$ are continuous, then $f(x)\pm g(x)$, $f(x)\times g(x)$ and $f(x)\div g(x)$ are continuous provided $g(x)\neq 0$. But what if $f(x)$ is continuous and $g(x)$ are discontinuous and we want to determine what type of function $f(x)\times g(x)$ is.
I have tried something out to prove that it will be discontinuous.
Let $h(x)=f(x)\times g(x)$.
Thus, $g(x)=\frac{h(x)}{f(x)}$, where $f(x)\neq 0$If $h(x)$ is continuous, then $g(x)$ is also continuous. But this is contrary to our assumption.
So, $h(x)$ must be discontinuous.
So, $f(x)\times g(x)$ must be discontinuous.
This is alright but when I asked my teacher he said that the multiplication can yield both sort of functions, continuous and discontinuous. So, what have I missed in my derivation?
Should I have considered something else as well or was my conclusion wrong?
Best Answer
You have proven that if $f(x)\neq 0$ and $g$ is not continuous, then $f(x)\cdot g(x)$ is not continuous. Your proof is correct.
However, if $f(x)$ can be allowed to have zero values, all bets are off. For example, if $f(x)=0$ for all values of $x$, then $f\cdot g$ will be continuous no matter what the function $g$ is.