Continuity of a Function

A function's graph that can be drawn without ever lifting the pen from the paper is, informally, a continuous function — but business mathematics needs a precise test, not just a picture, since a function's continuity determines whether the differentiation techniques of the next chapter can even be applied to it at a given point. This is exactly the test an Odisha CHSE +2 Business Mathematics & Statistics student is expected to apply carefully, rather than judge a function's continuity by eye alone.

Continuity at a point. A function $f$ is said to be continuous at $x = a$ if all three of the following conditions hold:

1. $f(a)$ is defined (the function has an actual value at $a$).
2. $\lim_{x \to a} f(x)$ exists (the LHL and RHL are equal — see the earlier section).
3. $\lim_{x \to a} f(x) = f(a)$ (the limit, once it exists, actually equals the function's value at that point).

If any one of these three conditions fails, $f$ is said to be discontinuous at $x = a$.

Continuity on an interval. A function is continuous on an open interval $(p, q)$ if it is continuous at every point of that interval; it is continuous on a closed interval $[p, q]$ if, in addition, it is continuous from the right at $p$ and from the left at $q$ (only one side needs checking at the endpoints, since there is no 'other side' within the interval to approach from).

Two ways continuity can fail. It helps to recognise that condition 2 and condition 3 fail for genuinely different reasons:

• Jump discontinuity — the LHL and RHL themselves are unequal, so condition 2 fails and the limit does not exist at all. The function's graph genuinely 'jumps' from one level to another at that point (Worked Example 1's piecewise function is exactly this case). …