Question

Discuss the continuity of the function

$straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell 1 half minus straight x space space comma space 0 less or equal than straight x less than 1 half end cell row cell 1 space space space space space space comma space straight x equals 1 half end cell row cell 3 over 2 minus straight x space space comma 1 half less than straight x less or equal than 1 end cell end table close at space straight x equals 1 half$

Solution

straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell 1 half minus straight x space space comma space 0 less or equal than straight x less than 1 half end cell row cell 1 space space space space space space comma space straight x equals 1 half end cell row cell 3 over 2 minus straight x space space comma 1 half less than straight x less or equal than 1 end cell end table close space Lt with straight x rightwards arrow 1 half to the power of minus below straight f left parenthesis straight x right parenthesis equals space Lt with straight x rightwards arrow 1 half to the power of minus below open parentheses 1 half minus straight x close parentheses equals space Lt with straight h rightwards arrow 0 below open curly brackets 1 half minus open parentheses 1 half minus straight h close parentheses close curly brackets space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets Put space straight x equals 1 half minus straight h comma space straight h greater than 0 space so space that space straight h rightwards arrow 0 space as space straight x rightwards arrow 1 half to the power of minus close square brackets space space space space space space space space space space space space space space space space space equals Lt with straight h rightwards arrow 0 below straight h equals 0 But space straight f open parentheses 1 half close parentheses equals 1 therefore space Lt with straight x rightwards arrow 1 half to the power of minus below straight f left parenthesis straight x right parenthesis not equal to straight f open parentheses 1 half close parentheses space space space space space space space space space space space space space space space space space space space space rightwards double arrow space straight f left parenthesis straight x right parenthesis space is space discontinous space at space straight x equals 1 half.

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