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

f (x )= \left\{\begin{matrix} \frac{|x|}{x} & if \: \: x \neq 0 \\ 0 & if \: \: x = 0 \end{matrix}\right.

 

Answers (1)

Given function is
f (x ) \left\{\begin{matrix} \frac{|x|}{x} & if \: \: x \neq 0 \\ 0 & if \: \: x = 0 \end{matrix}\right.
if x > 0 ,  f(x)=\frac{x}{x} = 1
if x < 0 , f(x)=\frac{-(x)}{x} = -1
given function is defined for every real number k 
Now,
case(i) k < 0 
f(k) = -1\\ \lim_{x\rightarrow k }f(x) = -1\\ \lim_{x\rightarrow k }f(x) = f(k)
Hence, given function is continuous for every value of k < 0
case(ii)  k > 0 
f(k) = 1\\ \lim_{x\rightarrow k }f(x) = 1\\ \lim_{x\rightarrow k }f(x) = f(k)
Hence, given function is continuous for every value of k > 0
case(iii)  x = 0
f(0) = 0\\ \lim_{x\rightarrow 0^- }f(x) = -1\\ \lim_{x\rightarrow 0^+}f(x) = 1\\ f(0) \neq R.H.L. \neq L.H.L.
Hence, 0 is the only point of discontinuity
 

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