Determine if the following function is continuous at $x = 1$ if $f (x) = 3 x - 5$ if $x \neq 1$ and $2$ if $x = 1$ .

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Solution

Function $f$ is defined at $x = 1$ since
$f (1) = 2$
${lim}_{x \to 1} f (x) = {lim}_{x \to 1} (3 x - 5) = 3 \times 1 - 5 = - 2$
But ${lim}_{x \to 1} f (x) \neq f (1)$
So, condition $2$ is not satisfied and $f$ is not continuous at $x = 1$

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