LetDiscuss continuity of f(x)

Question

Let
$straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell 1 minus straight x space space space comma space space 0 less or equal than straight x less than 1 half end cell row cell 0 space space space comma space space straight x equals 1 half end cell row cell space space space straight x space space space space space space comma space space 1 half less than straight x less than 1 end cell end table close$
Discuss continuity of f(x) $at space straight x equals 1 half$

Solution

We space have space straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell 1 minus straight x space space space comma space space 0 less or equal than straight x less than 1 half end cell row cell 0 space space space comma space space straight x equals 1 half end cell row cell space space space straight x space space space space space space comma space space 1 half less than straight x less 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 left parenthesis 1 minus straight x right parenthesis 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 equals Lt with straight h rightwards arrow 0 below open curly brackets 1 minus open parentheses 1 half minus straight h close parentheses close curly brackets equals Lt with straight h rightwards arrow 0 below open parentheses 1 half plus straight h close parentheses equals 1 half plus 0 equals 1 half Also space straight f open parentheses 1 half close parentheses equals 0 therefore space Lt with straight x rightwards arrow 1 half to the power of minus below space straight f left parenthesis straight x right parenthesis not equal to straight f open parentheses 1 half close parentheses therefore 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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