6. Find all points of discontinuity of f, where f is defined by

f (x) = \left\{\begin{matrix} 2x+3 & if x \leq 2 \\ 2x-3 & if x \geq 2 \end{matrix}\right.

 

Answers (1)

Given function is
f (x) = \left\{\begin{matrix} 2x+3 & if x \leq 2 \\ 2x-3 & if x \geq 2 \end{matrix}\right.
given function is defined for every real number k 
There are different cases for the given function
case(i)   k > 2
f(k) = 2k-3\\ \lim_{x\rightarrow k}f(x) = 2k-3\\ \lim_{x\rightarrow k}f(x) = f(k)
Hence, given function is continuous for each value of k > 2

case(ii)   k < 2
f(k) = 2k +3\\ \lim_{x\rightarrow k}f(x) = 2k+3\\ \lim_{x\rightarrow k}f(x) = f(k)
Hence, given function is continuous for each value of k < 2

case(iii)  x = 2

\lim_{x\rightarrow 2^-}f(x) = 2x+3 = 2\times 2 + 3 = 4 + 3 = 7\\ \lim_{x\rightarrow 2^+}f(x) = 2x-3 = 2\times 2-3 = 4-3 = 1
Right hand limit at x= 2 \neq Left hand limit at x = 2  
Therefore, x = 2 is the point of discontinuity

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