Question

Find all points of discontinuity of f, where f is defined by
$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight x to the power of 10 minus 1 comma space space if space straight x less or equal than 1 end cell row cell straight x squared comma space space space space space space space space space space space if space straight x greater than 1 end cell end table close$

Solution

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight x to the power of 10 minus 1 comma space space if space straight x less or equal than 1 end cell row cell straight x squared comma space space space space space space space space space space space if space straight x greater than 1 end cell end table close

Function/is defined at all points of the real line.
Let c be any real number.
Three cases arise :
Case I : c < 1

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 to the power of 10 minus 1 right parenthesis equals straight c to the power of 10 minus 1 space space space space space space space space space space space space space straight f left parenthesis straight c right parenthesis equals straight c to the power of 10 minus 1 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

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 straight x squared equals straight c squared space space space space space space space space space space space space space straight f left parenthesis straight c right parenthesis equals straight c squared 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 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 to the power of 10 minus 1 right parenthesis equals 1 to the power of 10 minus 1 equals 1 minus 1 equals 0 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 straight x squared equals left parenthesis 1 right parenthesis squared equals 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 discontinuous at x = 1

For Daily Free Study Material Join wiredfaculty