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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):}`
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Solution
f(x) = `{(|x|/x", if" x != 0),(0", if" x = 0):}`
`lim_(x -> 0^-)` f(x) = `lim_(x -> 0^-) abs x/x`
= `lim_(x -> 0^-) (-x)/x`
= `lim_(x -> 0^-)` (−1)
= −1
`lim_(x -> 0^+)` f(x) = `lim_(x -> 0^+) abs x/x`
= `lim_(x -> 0^+) x/x`
= `lim_(x -> 0^+)` (1)
= 1
Hence, f is not continuous at x = 0.
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