Q.Examine the continuity of $f(x) = x^3 + 2x^2 - x - 2$ at $x = -2$.

If two functions $f$ and $g$ are each already known to be continuous at a point $x=a$, then several natural combinations built from them are automatically continuous there too, without any further checking: their sum $f+g$, their difference $f-g$, a constant multiple $k\cdot f$ for any real $k$, their product $f\cdot g$, their quotient $f/g$ (provided $g(a)\ne0$, so division by zero is avoided), and either composite function $f[g(x)]$ or $g[f(x)]$. This algebra of continuity is the underlying reason why so many familiar functions turn out to be continuous everywhere they are defined: a polynomial is continuous because it is only ever built from the constant function and the identity function $x$ using sums, differences and products; a rational function is continuous away from the zeros of its denominator because it is simply a quotient of two polynomials; and more elaborate expressions such as $\sin(x^2+1)$ or $\log(\cos x)$ are continuous wherever they are defined because they are composites of individually continuous pieces. Recognising a function as a combination of already-continuous building blocks is usually much faster than returning to the three-condition definition every single time.