Q.Examine the continuity of $f(x) = \sin x$, for $x \le \dfrac{\pi}{4}$, $= \cos x$, for $x > \dfrac{\pi}{4}$, at $x = \dfrac{\pi}{4}$.

When a function is built from two (or more) different formulas that switch over at a particular point, its continuity there has to be tested one side at a time rather than all at once. A function is continuous from the right at $x=a$ if $\lim_{x\to a^+} f(x)=f(a)$, and continuous from the left at $x=a$ if $\lim_{x\to a^-} f(x)=f(a)$. Only when both of these one-sided checks succeed together — right-hand limit equals $f(a)$, and left-hand limit equals $f(a)$ — is the function genuinely continuous at that junction point, because then the two one-sided limits automatically agree with each other and with $f(a)$. This is the standard method for handling piecewise-defined functions: evaluate $f(a)$ from whichever piece includes the point itself, then compute the left-hand limit using the formula valid just below $a$ and the right-hand limit using the formula valid just above $a$, and check all three numbers coincide. If the one-sided limits exist but disagree with each other, the function has a jump discontinuity there instead.