Q.Discuss the continuity of the following function at the point indicated against it: $f(x) = \dfrac{e^{1/x}-1}{e^{1/x}+1}$, for $x \ne 0$, $= 1$, for $x = 0$, at $x = 0$.

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 …