f{x}={x/|x|, x not equal to 0 and 1,x=0} Is the f{x} is continuous

Asked by jayshreechoudhury | 17th Aug, 2020, 11:22: AM

Expert Answer:

To check the continuity of f(x) at x=0
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 vertical line straight x vertical line end fraction space comma space straight x not equal to 0 end cell row cell 1 space space space space comma space straight x equals 0 end cell end table close equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator straight x over denominator negative straight x end fraction comma space straight x less than 0 end cell row cell straight x over straight x space space comma space straight x greater than 0 end cell row cell 1 space space space comma space straight x equals 0 end cell end table close equals open curly brackets table attributes columnalign left end attributes row cell negative 1 comma space straight x less than 0 end cell row cell 1 space space space comma space straight x greater than 0 end cell row cell 1 space space space comma space straight x equals 0 end cell end table close
rightwards double arrow straight f left parenthesis straight x right parenthesis equals equals open curly brackets table attributes columnalign left end attributes row cell negative 1 comma space straight x less than 0 end cell row cell 1 space space space comma space straight x greater than 0 end cell row cell 1 space space space comma space straight x equals 0 end cell end table close
Left space hand space limit equals limit as straight x rightwards arrow 0 to the power of minus of straight f left parenthesis straight x right parenthesis equals limit as straight x rightwards arrow 0 of minus 1 equals negative 1
Right space hand space limit equals limit as straight x rightwards arrow 0 to the power of plus of straight f left parenthesis straight x right parenthesis equals limit as straight x rightwards arrow 0 of 1 equals 1
straight f left parenthesis straight x right parenthesis equals 1 space at space straight x equals 0
limit as straight x rightwards arrow 0 to the power of minus of straight f left parenthesis straight x right parenthesis not equal to limit as straight x rightwards arrow 0 to the power of plus of straight f left parenthesis straight x right parenthesis
Hence comma space straight f left parenthesis straight x right parenthesis space is space not space continuos space at space straight x equals 0.

Answered by Renu Varma | 17th Aug, 2020, 11:34: AM