Question

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

$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator straight x over denominator open vertical bar straight x close vertical bar end fraction comma space if space straight x less than 0 end cell row cell space minus 1 comma space space if space straight x greater or equal than 0 end cell end table close$

Solution

Here space straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator straight x over denominator open vertical bar straight x close vertical bar end fraction comma space if space straight x less than 0 end cell row cell space minus 1 comma space space if space straight x greater or equal than 0 end cell end table close

Function f is defined for all points of the real line.
Let c be any real number.
Three cases arise :
Case I: c < 0

space space space space space space space space space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight c below fraction numerator straight x over denominator open vertical bar straight x close vertical bar end fraction equals Lt with straight x rightwards arrow straight c below open parentheses fraction numerator straight x over denominator negative straight x end fraction close parentheses equals Lt with straight x rightwards arrow straight c below left parenthesis negative 1 right parenthesis equals negative 1 Also space space space space space space space space straight f left parenthesis straight c right parenthesis equals fraction numerator straight c over denominator open vertical bar straight c close vertical bar end fraction equals fraction numerator straight c over denominator negative straight c end fraction equals negative 1 therefore space space space space space space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis straight c right parenthesis

∴ f is continuous at all points x < 0.
Case II; c > 0

space space space space space space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight c below fraction numerator straight x over denominator open vertical bar straight x close vertical bar end fraction equals Lt with straight x rightwards arrow straight c below open parentheses straight x over straight x close parentheses equals Lt with straight x rightwards arrow straight c below left parenthesis 1 right parenthesis equals 1 Also space space space space space straight f left parenthesis straight c right parenthesis equals fraction numerator straight c over denominator open vertical bar straight c close vertical bar end fraction equals straight c over straight c equals 1 therefore space space space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis straight c right parenthesis

∴ f is continuous at all points x > 0.
Case III; c = 0

space space space space space space space space space Lt with straight x rightwards arrow straight c to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 0 to the power of minus below fraction numerator straight x over denominator open vertical bar straight x close vertical bar end fraction equals Lt with straight x rightwards arrow 0 to the power of minus below open parentheses fraction numerator straight x over denominator negative straight x end fraction close parentheses equals Lt with straight x rightwards arrow 0 to the power of minus below left parenthesis negative 1 right parenthesis equals negative 1 space space space space space space space space space Lt with straight x rightwards arrow straight c to the power of plus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 0 to the power of plus below fraction numerator straight x over denominator open vertical bar straight x close vertical bar end fraction equals Lt with straight x rightwards arrow 0 to the power of plus below open parentheses straight x over straight x close parentheses equals Lt with straight x rightwards arrow 0 to the power of plus below left parenthesis 1 right parenthesis equals 1 therefore space space space space space space Lt with straight x rightwards arrow straight c to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight c to the power of plus below straight f left parenthesis straight x right parenthesis

∴ f is discontinuous at x = 0.

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