Prove that the identity function on real numbers given by $f(x) = x$ is continuous at every real number.
(N/A) The identity function $f(x) = x$ is defined for all real numbers $x \in \mathbb{R}$.
To check for continuity at any arbitrary real number $c$,we evaluate the limit of the function as $x$ approaches $c$:
$\lim_{x \to c} f(x) = \lim_{x \to c} x = c$.
Also,the value of the function at $x = c$ is $f(c) = c$.
Since $\lim_{x \to c} f(x) = f(c) = c$,the condition for continuity is satisfied for every real number $c$.
Therefore,the identity function $f(x) = x$ is continuous at every real number.
To check for continuity at any arbitrary real number $c$,we evaluate the limit of the function as $x$ approaches $c$:
$\lim_{x \to c} f(x) = \lim_{x \to c} x = c$.
Also,the value of the function at $x = c$ is $f(c) = c$.
Since $\lim_{x \to c} f(x) = f(c) = c$,the condition for continuity is satisfied for every real number $c$.
Therefore,the identity function $f(x) = x$ is continuous at every real number.
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