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Continuity And Differentiability

Question

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Examine the following functions for continuity :
$straight f left parenthesis straight x right parenthesis equals fraction numerator 1 over denominator straight x minus 5 end fraction$

Solution

Here space straight f left parenthesis straight x right parenthesis equals fraction numerator 1 over denominator straight x minus 5 end fraction

For f to be defined,
x – 5 ≠ 0 i.e. x ≠ 5
∴D_f = Set of real number except 5 = R - { 5}
Let c ≠ 5 be any real number.

Also space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis minus Lt with straight x rightwards arrow straight c below open parentheses fraction numerator 1 over denominator straight x minus 5 end fraction close parentheses equals fraction numerator 1 over denominator straight c minus 5 end fraction 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 x = c.
But c ≠ 5 is any real number
∴ f is continuous at every real number c ∈ D
∴ f is continuous function.

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