Question

Find the points of discontinuity, if any, of the following functions: 

\[f\left( x \right) = \begin{cases}\left| x \right| + 3 , & \text{ if } x \leq - 3 \\ - 2x , & \text { if }  - 3 < x < 3 \\ 6x + 2 , & \text{ if }  x > 3\end{cases}\]

Answer 1

At 

\[x \leq - 3\], we have

\[f\left( x \right) = \left| x \right| + 3\]

Since modulus function and constant function are continuous, 

\[f\left( x \right) = \left| x \right| + 3\]  is continuous for each \[x \leq - 3\]

At 

\[- 3 < x < 3\]  we have

 

\[f\left( x \right) = - 2x\]

 

Since polynomial function is continuous and constant function is continuous,

 

\[f\left( x \right) = - 2x\] is continuous for each \[- 3 < x < 3\]

At 

 

\[x > 3\] , we have

 

\[f\left( x \right) = 6x + 2\]

 

Since polynomial function is continuous and constant function is continuous, 

 

\[f\left( x \right) = 6x + 2\]  is continuous for each \[x > 3\] .

 

Now, we check the continuity of the function at the point  \[x = 3\] .

We have 

(LHL at x=3) = \[\lim_{x \to 3^-} f\left( x \right) = \lim_{h \to 0} f\left( 3 - h \right) = \lim_{h \to 0} - 2\left( 3 - h \right) = - 6\]

(RHL at x=3) =  \[\lim_{x \to 3^+} f\left( x \right) = \lim_{h \to 0} f\left( 3 + h \right) = \lim_{h \to 0} 6\left( 3 + h \right) + 2 = 20\]

\[\Rightarrow \lim_{x \to 3^-} f\left( x \right) \neq \lim_{x \to 3^+} f\left( x \right)\]

 

Hence, the only point of discontinuity of the given function is \[x = 3\].