Prove that the identity function on real numbers given by f(x) = … - Vidyadip

Question

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

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Answer

The function is clearly defined at every point and f(c) = c for every real number c. Also,
$\mathop {\lim }\limits_{x \to c} f(x) = \mathop {\lim }\limits_{x \to c} $ x = c
Thus $\mathop {\lim }\limits_{x \to c} f(x)$ = c = f(c)
and hence the function is continuous at every real number c $\in $ R.

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