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Is the function defined by $f(x)=\left\{\begin{array}{l}x+5, \text { if } x \leq 1 \\ x-5, \text { if } x>1\end{array}\right.$ a continuous function?
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$\begin{aligned}
&\text { At } x=1, \text { L.H.L. }=\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1^{-}}(x+5)=6 \\
&\text { R.H.L. }=\lim _{x \rightarrow 1^{+}} f(x)=\lim _{x \rightarrow 1^{+}}(x-5)=-4 \\
&\qquad f(1)=1+5=6 \\
&f(1)=\text { L.H.L. } \neq \text { R.H.L. } \\
&\Rightarrow \text { f is not continuous at } x=1
\end{aligned}$
At $x=c < 1, \lim _{x \rightarrow c}(x+5)=c+5=f(c)$
At $x=c>1, \lim _{x \rightarrow c}(x-5)=c-5=f(c)$
$\therefore \mathrm{f}$ is continuous at all points $\mathrm{x} \in \mathrm{R}$ except $\mathrm{x}=1$.
&\text { At } x=1, \text { L.H.L. }=\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1^{-}}(x+5)=6 \\
&\text { R.H.L. }=\lim _{x \rightarrow 1^{+}} f(x)=\lim _{x \rightarrow 1^{+}}(x-5)=-4 \\
&\qquad f(1)=1+5=6 \\
&f(1)=\text { L.H.L. } \neq \text { R.H.L. } \\
&\Rightarrow \text { f is not continuous at } x=1
\end{aligned}$
At $x=c < 1, \lim _{x \rightarrow c}(x+5)=c+5=f(c)$
At $x=c>1, \lim _{x \rightarrow c}(x-5)=c-5=f(c)$
$\therefore \mathrm{f}$ is continuous at all points $\mathrm{x} \in \mathrm{R}$ except $\mathrm{x}=1$.
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