Q.Identify discontinuities for the following function as either a jump or a removable discontinuity: $f(x) = x^2-3x-2$, for $x < -3$, $= 3+8x$, for $x > -3$.

A jump discontinuity occurs at a point $x=a$ when both the left-hand limit $\lim_{x\to a^-} f(x)$ and the right-hand limit $\lim_{x\to a^+} f(x)$ exist as finite numbers, but the two numbers are different from one another, so that $\lim_{x\to a^-} f(x)\ne\lim_{x\to a^+} f(x)$. Because both one-sided limits are perfectly well-defined individually, the graph does not blow up or vanish near $a$ — it simply steps abruptly from one finite height to a different finite height as $x$ crosses over $a$, exactly like a staircase. This is fundamentally different from a removable discontinuity: since the two sides genuinely disagree, there is no single value that could be assigned to $f(a)$ to repair the function into something continuous — redefining the point can fix at most one s …