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  • State True or False for the statements If f is continuous on its domain D, then |f| is also continuous on D.

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

Answers (1)

Solution

True.

Given that, f is continuous on its domain D.

Let a be an arbitrary real number in D. Then f is continuous at a.

\lim _{x \rightarrow a} f(x)=f(a)
Now,
\lim _{x \rightarrow a}|f|(x)=\lim _{x \rightarrow a}|f(x)|_{[\because|f|(x)=\mid f(x) L]}
\lim _{x \rightarrow a}|f|(x)=\left|\lim _{x \rightarrow a} f(x)\right| \\ \lim _{x \rightarrow a}|f|(x)=|f(a)|=|f|(a)
If |f| is continuous at x=a.
since a is an arbitrary point in D. Therefore |f| is continuous in D.

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