Q.Select the correct answer from the given alternatives. If $f(x) = ax^2+bx+1$, for $|x-1| \ge 3$ and $= 4x+5$, for $-2 < x < 4$, is continuous everywhere then, (A) $a=\dfrac12, b=3$ (B) $a=-\dfrac12, b=-3$ (C) $a=-\dfrac12, b=3$ (D) $a=\dfrac12, b=-3$

Continuity can be extended from a single point to an entire interval of points. A function $f$ is continuous on an open interval $(a,b)$ if it is continuous at every individual point inside that interval, with no exception. For a half-open interval such as $[a,b)$, $f$ must additionally be continuous from the right at the included endpoint $a$ (i.e. $\lim_{x\to a^+} f(x)=f(a)$); similarly, for $(a,b]$, $f$ must be continuous from the left at $b$. A function is continuous on a full closed interval $[a,b]$ precisely when it is continuous at every point of the open interior $(a,b)$, continuous from the right at $a$, and continuous from the left at $b$. This distinction matters in practice for functions like the greatest-integer function, or any piecewise function with several joins, where continuity has to be checked …