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Q)

True or False: If f.g is continuous at $x=a$,then f and g are separately continuous at $x=a.$

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A)
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  • A function is said to be continuous on an open interval $(a,b)$ if it is continuous at every point on the interval $(a,b)$.$\mid f\mid$ is defined as $\mid f\mid(x)=\mid f(x)\mid$
If $f.g$ is continuous at $x=0$ then $f$ and $g$ are separately continuous at $x=0$.
Let us consider $f(x)=\large\frac{1}{x}$ and $g(x)=x$.
$f.g=1$
This is continuous at $x=0$
But even though $g$ is continuous,$f$ is discontinuous at $x=0$
Hence it is a False statement.
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