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81. If f is continuous and g is a discontinuous function then f + g is discontinuous function.

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82. Examine the continuity 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 sin space straight x minus cos space straight x comma space space space if space straight x not equal to 0 end cell row cell space space space space space space minus 1 space space space space space space space space space comma space space space if space straight x equals 0 end cell end table close

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83. Find all points of discontinuity of f where

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

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84. Let f be defined by

straight f left parenthesis straight x right parenthesis open curly brackets table attributes columnalign left end attributes row cell space space space space space straight x squared over straight x space space space space space space space space space space space space comma space 0 space less or equal than straight x less or equal than 1 end cell row cell 2 straight x squared minus 3 straight x plus 3 over 2 comma space 1 less than straight x less or equal than 2 end cell end table close

Show that f is continous at x=1.

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85. Is the function f defined by

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight x comma space if space straight x less or equal than 1 end cell row cell 5 comma space if space straight x greater than 1 end cell end table close

continous at x=0? At x=1? At x=2?

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86. Which of the following functions are continuous at the indicated points ?

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight x space space space comma space if space straight x less than 0 end cell row cell straight x squared space space comma space if space straight x greater than 0 end cell end table close

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87. Which of the following functions are continuous at the indicated points?

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell space space 5 straight x minus 4 space space space comma space if space straight x less or equal than 1 end cell row cell 4 straight x squared minus 3 straight x space space comma space if space straight x greater than 1 end cell end table close

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88. Which of the following functions are continuous at the indicated points ?

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

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89. Which of the following functions are continuous at the indicated points ?

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell 2 straight x minus 1 space space space comma space space space if space straight x less than 0 end cell row cell 2 straight x plus 6 space space space comma space space space if space straight x greater than 1 end cell end table close at space straight x equals 0

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90. Which of the following functions are continuous at the indicated points ?

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell 2 straight x minus 1 space space space comma space space space if space space space straight x less than 2 end cell row cell space space space space space space 3 over 2 space space space space comma space space space if space space space straight x greater or equal than 2 end cell end table close at space straight x equals 2

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