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Q)

True or False: If f is continuous on its domain D ,then | f | is also continuous on D.

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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 $\mid f\mid(x)=\mid f(x)\mid$
Let $'a'$ be an arbitrary real number in domain $D$.Then $f$ is continuous at $a$.
Therefore $\lim\limits_{\large x\to a}f(x)=f(a)$
Now $\lim\limits_{\large x\to a}\mid f\mid(x)=\lim\limits_{\large x\to a}\mid f(x)\mid$
$\Rightarrow \mid f(a)\mid=\mid f\mid(a)$
Therefore $\mid f\mid$ is continuous at $x=a$
Therefore it is a True statement.
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