Question

Discuss the continuity of the function

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

Solution

Here space straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell space space space space space space space straight x space space comma space 0 less or equal than straight x less than 1 half end cell row cell straight x over 2 space comma space straight x equals 1 half end cell row cell space 1 minus straight x space space comma space 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 Lt with straight x rightwards arrow 1 half to the power of minus below left parenthesis straight x right parenthesis equals Lt with straight h rightwards arrow 0 below open parentheses 1 half minus straight h close parentheses 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 space space equals 1 half minus 0 equals 1 half space Lt with straight x rightwards arrow 1 half to the power of plus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 1 half to the power of plus below left parenthesis 1 minus straight x right parenthesis equals Lt with straight h rightwards arrow 0 below open curly brackets 1 minus open parentheses 1 half plus 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 open square brackets Put space straight x equals 1 half plus 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 plus close square brackets space space 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 open parentheses 1 half minus straight h close parentheses equals 1 half minus 0 equals 1 half Also space straight f open parentheses 1 half close parentheses equals 1 half therefore space Lt with straight x rightwards arrow 1 half to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 1 half to the power of plus below straight f left parenthesis straight x right parenthesis equals straight f open parentheses 1 half close parentheses equals 1 half comma space which space is space finite. therefore space straight f left parenthesis straight x right parenthesis space is space continous space at space straight x equals 1 half.

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