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Continuity And Differentiability

Question

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Discuss the continuity of the function f defined by
$straight f left parenthesis straight x right parenthesis equals 1 over straight x comma space straight x not equal to 0$

Solution

Here space straight f left parenthesis straight x right parenthesis equals 1 over straight x comma space straight x not equal to 0

Function f is defined for all real numbers except 0.
Let c ≠ 0 be any real number.

therefore space straight f left parenthesis straight c right parenthesis equals 1 over straight c Also space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight c below 1 over straight c equals 1 over straight c therefore space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis straight c right parenthesis

∴ f is continuous at c
But c ≠ 0 is any real number
∴ f is continuous at every real number except 0.
∴ f is a continuous function at its domain.

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