Continuity of a Function at a Point

A function's continuity at a point is a direct application of the LHL = RHL idea from Section 1, with one extra condition added: the common limit value must also match the function's actual value at that point.

Definition. A function $f(x)$ is said to be continuous at $x=a$ if all three of the following hold:

1. $f(a)$ is defined (the function has a real value at $x=a$).
2. $\displaystyle\lim_{x\to a}f(x)$ exists, i.e. $\displaystyle\lim_{x\to a^{-}}f(x)=\lim_{x\to a^{+}}f(x)$.
3. $\displaystyle\lim_{x\to a}f(x)=f(a)$ — the limit equals the actual function value.

If any one of the three conditions fails, $f$ is discontinuous at $x=a$. Two discontinuity patterns matter most for the Gujarat board Std 12 Statistics syllabus and for the business examples in Section 5:

• A jump discontinuity — the LHL and RHL both exist but are unequal (the graph genuinely "jumps" from one value to another at $x=a$). This is exactly the pattern behind slab-based pricing and tax structures.
• A removable discontinuity — the LHL and RHL are equal to each other but do not match $f(a)$ (or $f(a)$ is simply undefined), so the limit exists but the function's own definition at that single point spoils continuity. …