Discuss the continuity of the function $f$ given by $f(x)=x^{3}+x^{2}-1$.

Step 1: Identify the function type

The given function is $f(x) = x^3 + x^2 - 1$. This is a polynomial function, as it consists of terms involving non-negative integer powers of $x$ combined with addition and subtraction.

Step 2: Recall continuity property of polynomials

A fundamental property of polynomial functions is that they are continuous for all real numbers. This means that for any real number $c$, the limit of the function as $x$ approaches $c$ exists and is equal to the function's value at $c$, i.e., $\lim_{x \to c} f(x) = f(c)$.

Step 3: Conclusion on continuity

Since $f(x) = x^3 + x^2 - 1$ is a polynomial function, it is continuous for all real numbers. There are no points where the function is undefined or where its graph would have a break.

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