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Question
Find all points of discontinuity of f, where f is defined by `f(x) = {(|x|/x , if x != 0),(0, if x = 0):}`
Solution
`f(x) = abs x/x, if x ne 0` is not equal to 0, if x = 0
`lim_(x -> 0^-) f(x) = lim_(x -> 0^-) abs x/x`
= `lim_(h -> 0) abs (0 - h)/(0 - h)`
= `lim_(h -> 0) h/-h = -1`
`h -> 0`
`lim_(x -> 0^+) f(x) = lim_(x -> 0^+) abs x/x`
= `lim_(h -> 0) abs (0 + h)/(0 + h)`
= `lim_(h -> 0) h/h = 1`
Hence, f is not continuous at x = 0.
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