Question

Find all the points of discontinuity of the function f defined by

$straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell straight x plus 2 comma space if space straight x less than 1 end cell row cell space space space space 0 comma space space space if space straight x equals 1 end cell row cell straight x minus 2 comma space if space straight x greater than 1 end cell end table close$

Solution

Here space straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell straight x plus 2 comma space if space straight x less than 1 end cell row cell space space space space 0 comma space space space if space straight x equals 1 end cell row cell straight x minus 2 comma space if space straight x greater than 1 end cell end table close

Function f is defined for ail points of the real line.
Let c be any real number.
Three cases arise :
Case I : c < 1

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 left parenthesis straight x plus 2 right parenthesis equals straight c plus 2 space space space space space space space space space space space space straight f left parenthesis straight c right parenthesis equals straight c plus 2 therefore 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 < 1.
Case II : c > 1

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 left parenthesis straight x minus 2 right parenthesis equals straight c minus 2 space space space space space space space space space space space space straight f left parenthesis straight c right parenthesis equals straight c minus 2 therefore 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 > 1.
Case III : c = 1

space space space space space Lt with straight x rightwards arrow 1 to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 1 to the power of minus below left parenthesis straight x plus 2 right parenthesis equals 1 plus 2 equals 3 space space space space Lt with straight x rightwards arrow 1 to the power of plus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 1 to the power of plus below left parenthesis straight x minus 2 right parenthesis equals 1 minus 2 equals negative 1 therefore Lt with straight x rightwards arrow 1 to the power of minus below straight f left parenthesis straight x right parenthesis not equal to Lt with straight x rightwards arrow 1 to the power of plus below straight f left parenthesis straight x right parenthesis

∴ f is not continuous at x = 1
∴ x = 1 is the only point of discontinuity of f

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