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Q)

True or False: The composition of two continuous functions is a continuous function.

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A)
Toolbox:
  • 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 $\mid f\mid(x)=\mid f(x)\mid$
Step 1:
Let $f$ and $g$ be two real functions such that $gof$ exists.
Then Range(f) $\leq$ Domain(g)
Let $'a'$ be an arbitrary point in the domain of $f$.
Then $a\in$ Domain (f) and $f(a)\in $ Domain (g).
$\Rightarrow f$ is continuous at $x=a$ and $g$ is continuous at $f(a)$.
$\Rightarrow \lim\limits_{\large x\to a}f(x)=f(a)$ and $\lim\limits_{\large y\to a}g(y)=g(f(a))$
$\Rightarrow \lim\limits_{\large x\to a}f(x)=f(a)$ and $\lim\limits_{\large f(x)\to f(a)}g(f(x))=g(f(a))$
Step 2:
Where $y=f(x)$
$\Rightarrow \lim\limits_{\large x\to a}g(f(x))=g(f(a))$
$x\to a\Rightarrow f(x)\to f(a)$
$\Rightarrow \lim\limits_{\large x\to a}gof(x)=gof(a)$
$gof$ is continuous at $x=a$
Hence it is a True statement.
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