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I'll state the question from my textbook below:

State whether the below statement is true or false:

If $f\cdot g$ is continuous at $x=a$, then $f$ and $g$ are both continuous at $x=a$.

The textbook says that it is false.

I tried thinking of a very basic example considering $f(x) = x$ and $g(x) = \frac 1x$. But then I recalled that $p(x) = \frac xx$ and $q(x) = 1$ are different functions. So, it didn't work as a counter example.

So, can someone please provide a counter example? Or is the statement true, just as I think?

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It is false. As example You can take $f=x$, $g=\theta(x)$ (Heaviside function)

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Take $f = 1_{[a,\infty)}$ and $g = 1_{(-\infty,a)}$. Then $fg = 0$, but none are continuous at $a$.

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  • $\begingroup$ What does the subscript after $1$ denote? $\endgroup$ – SamInuyasha ANMF Apr 26 '18 at 14:34
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    $\begingroup$ @SamInuyashaANMF The function $1_{[a, b]}$ refers to the function which is equal to $1$ in $[a, b]$, and $0$ elsewhere. $\endgroup$ – Clarinetist Apr 26 '18 at 14:35

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