Question

Prove that the identity function on real numbers given by f(x) = x is continuous at every real number.

Solution

Here f(x) = x
Function f is defined for all real numbers
Let c be any real number.
$therefore space straight f left parenthesis straight c right parenthesis equals straight c Also space Lt with straight x rightwards arrow straight c below space straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight c below space straight x equals straight c therefore space Lt with straight x rightwards arrow straight c below space straight f left parenthesis straight x right parenthesis equals space straight f left parenthesis straight c right parenthesis$
∴ f is continuous at x = c
But c is any real number.
∴ f is continuous at every real number.

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Continuity And Differentiability

Prove that the identity function on real numbers given by f(x) = x is continuous at every real number.

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