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Find all points of discontinuity of $f$ : where $f$ is defined by: $f(x)=\left\{\begin{array}{lll}\frac{x}{|x|}, & \text { if } & x<0 \\ -1, & \text { if } & x \geq 0\end{array}\right.$
Find all points of discontinuity of $f$ : where $f$ is defined by: $f(x)=\left\{\begin{array}{lll}\frac{x}{|x|}, & \text { if } & x<0 \\ -1, & \text { if } & x \geq 0\end{array}\right.$
CBSE | Class 12 | Continuity and Differentiability | Exercise 5.1 | Maths | NCERT
Find all points of discontinuity of $f$ : where $f$ is defined by: $f(x)=\left\{\begin{array}{lll}\frac{x}{|x|}, & \text { if } & x<0 \\ -1, & \text { if } & x \geq 0\end{array}\right.$

Solution:

The function provided is $f(x)=\left\{\begin{array}{lll}\frac{x}{|x|}, & \text { if } & x<0 \\ -1, & \text { if } & x \geq 0\end{array}\right.$

At $x=0$,
Left Hand Limit $=\lim _{x \rightarrow 0^{-}} \frac{x}{|x|}=-1$ And $f(0)=-1$

Right Hand Limit $=\lim _{x \rightarrow 0^{f}} f(x)=-1$

Since, L.H.L.$=$ R.H.L. $=f(0)$

As a result, $f(x)$ is a continuous function.

Now, for $x=c<0$ lim $_{x \rightarrow-^{-}} \frac{x}{|x|}=-1=f(c)$

Then, $\lim _{x \rightarrow-}=f(x)$

As a result, $f(x)$ is a continuous at $x=c<0$

Now, for $x=c>0 \lim _{x \rightarrow c^{-}} f(x)=1=f(c)$

As a result, $f(x)$ is a continuous at $x=c>0$

Hence, at all points of its domain, the function is continuous.

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