Question

Examine the continuity of f , where f is defined by
$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell sin space straight x minus cos space straight x comma space space space if space straight x not equal to 0 end cell row cell space space space space space space minus 1 space space space space space space space space space comma space space space if space straight x equals 0 end cell end table close$

Solution

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

D_f = R
Let a be any real number.
Two cases arise :
Case I : a ≠ 0

space Lt with straight x rightwards arrow straight a below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight a below left parenthesis sin space straight x minus cos space straight x right parenthesis 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 left square bracket Put space straight x equals straight a 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 straight a right square bracket 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 sin left parenthesis straight a plus straight h right parenthesis minus cos left parenthesis straight a plus straight h right parenthesis close curly brackets 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 left parenthesis sin space straight a space cos space straight h plus cos space straight a space cos space straight h right parenthesis minus left parenthesis cos space straight a space cos space straight h minus sin space straight a space sin space straight h right parenthesis close curly brackets space space space space space space space space space space space space space space space equals open curly brackets left parenthesis sin space straight a.1 plus cos space straight a.0 right parenthesis minus left parenthesis cos space straight a.1 minus sin space straight a.0 right parenthesis close curly brackets equals sin space straight a minus cos space straight a Also space straight f left parenthesis straight a right parenthesis equals sin space straight a minus cos space straight a therefore space Lt with straight x rightwards arrow straight a below straight f left parenthesis straight x right parenthesis equals sin space straight a minus cos space straight a

∴ f is continuous at x = a ≠ 0.

Case II : a = 0
$therefore space Lt with straight x rightwards arrow 0 to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 0 to the power of minus below left parenthesis sin space straight x minus cos space straight x right parenthesis 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 left square bracket Put space straight x equals 0 minus straight h comma straight h greater than 0 space so space that space straight h rightwards arrow 0 space as space straight x rightwards arrow 0 to the power of minus right square bracket space 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 curly brackets sin left parenthesis negative straight h right parenthesis minus cos left parenthesis negative straight h right parenthesis close curly brackets equals Lt with straight h rightwards arrow 0 below left parenthesis negative sin space straight h minus cos space straight h right parenthesis space space space space space space space space space space space space space space space space space space space space equals negative sin space 0 minus cos space 0 equals negative 0 minus 1 equals negative 1 space space space space Lt with straight x rightwards arrow 0 to the power of plus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 0 to the power of plus below left parenthesis sin space straight x minus space cos space straight x right parenthesis 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 left square bracket Put space straight x equals 0 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 0 to the power of plus right square bracket space 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 left parenthesis sin space straight h minus cos space straight h right parenthesis equals 0 minus 1 equals negative 1 Also space space space space space straight f left parenthesis 0 right parenthesis equals sin space 0 minus cos space 0 equals 0 minus 1 therefore space Lt with straight x rightwards arrow 0 to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 0 to the power of plus below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis 0 right parenthesis$
∴ f is continuous at x = a = 0
Thus, f is continuous at all points of its domain.

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