Q.Activity: Let $f(x) = ax + b$ (where $a$ and $b$ are unknown), for $x < 1$, $= x^2 + 5$, for $x \ge 1$. Find the values of $a$ and $b$, so that $f(x)$ is continuous at $x = 1$. (Fig. 8.11)

A function $f(x)$ is continuous at a point $x=a$ when three separate requirements all hold together: the function must actually be defined at $a$ (and on a whole open interval around it), the limit $\lim_{x\to a} f(x)$ must exist as a single finite number, and that limiting value must be exactly equal to $f(a)$ itself. If any one of these three fails — the value is missing, the two-sided limit does not exist, or the limit disagrees with the assigned value — the function is discontinuous at that point. An equivalent way to state the third condition is $\lim_{h\to0}[f(a+h)-f(a)]=0$, which says that as the input is nudged away from $a$ by a shrinking amount $h$, the resulting change in output must also shrink to zero, ruling out any sudden jump rig …