Discuss the continuity of the function f given by f(x) = x^3 + x^2 -…

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

Clearly $f$ is defined at every real number $c$ and its value at $c$ is $c^3 + c^2 - 1$. We also know that

$\lim\limits_{x \to c} f(x) = \lim\limits_{x \to c} (x^3 + x^2 - 1) = c^3 + c^2 - 1$

Thus $\lim\limits_{x \to c} f(x) = f(c)$, and hence $f$ is continuous at every real number. This means $f$ is a continuous function.