Check the points where the constant function f(x) = k is continuous.

The correct answer is Option (3) → Continuous at every real number. ##

The function is defined at all real numbers and by definition, its value at any real number equals $k$. Let $c$ be any real number. Then

$\lim\limits_{x \to c} f(x) = \lim\limits_{x \to c} k = k$

Since $f(c) = k = \lim\limits_{x \to c} f(x)$ for any real number $c$, the function $f$ is continuous at every real number.