Examine the following function for continuity: $f(x) = x - 5$.
(N/A) The given function is $f(x) = x - 5$.
It is evident that $f$ is defined at every real number $k$ and its value at $k$ is $f(k) = k - 5$.
We calculate the limit of the function as $x$ approaches $k$:
$\lim_{x \to k} f(x) = \lim_{x \to k} (x - 5) = k - 5$.
Since $\lim_{x \to k} f(x) = f(k)$,the function $f$ is continuous at every real number $k$.
Therefore,$f(x) = x - 5$ is a continuous function.
It is evident that $f$ is defined at every real number $k$ and its value at $k$ is $f(k) = k - 5$.
We calculate the limit of the function as $x$ approaches $k$:
$\lim_{x \to k} f(x) = \lim_{x \to k} (x - 5) = k - 5$.
Since $\lim_{x \to k} f(x) = f(k)$,the function $f$ is continuous at every real number $k$.
Therefore,$f(x) = x - 5$ is a continuous function.
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Statement $2$: $A$ function $f: R \to R$ is discontinuous at $x_0$ if and only if $\lim_{x \to x_0} f(x)$ exists and $\lim_{x \to x_0} f(x) \neq f(x_0)$.
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