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Let $f$ be a function such that $f(xy)=f(x)f(y^3)$ for all $x$ and $y$. If $f(x)$ is continuous at $x=1$, show that $f(x)$ is continuous at all $x$ except at $x=0$.

My work: for continuity $f(x+h)=f(x)$, where $h$ is a small increment in $x$, I am able to get $f(x+h)=f(x)f(1)$. I am unable to calculate $f(1)$. Please help

Klangen
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Siddhartha Raja
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1 Answers1

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You got that $f(xy)=f(x)f(y^3)$ so $f(1)=f(1)f(1)$ hence $f(1)=1$ or $f(1)=0$. If $f(1)=0$ then $f(x)=0$ for all $x$. If $f(1)=1$ then $f(x+h)=f(x)$ for all $h$, so $f$ is constant function.

maciek97x
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