Mathematics and Statistics · Ch 8 — Continuity

Continuity of a Function at a Point

Informally, a function is continuous at a point if its graph can be drawn through that point without lifting the pen — no break, no hole, and no sudden jump. To turn this picture into a precise, testable rule, recall from the Limits chapter that a function's limit at a point describes the value the function approaches, while $f(a)$ is the value the function actually takes there. Continuity is exactly the statement that these two agree.

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

$f(a)$ is defined (the point $a$ lies in the domain of $f$ );
$\displaystyle\lim_{x \to a} f(x)$ exists (the left-hand limit and right-hand limit are both finite and equal to each other);
$\displaystyle\lim_{x \to a} f(x) = f(a)$ (the value approached equals the value taken).

If even one of these three conditions fails, $f$ is discontinuous at $x = a$ . The three-condition test is the single most important tool of this chapter — every worked example below checks these conditions one by one, in order.

Left-hand and right-hand limits. The second condition unpacks into the two one-sided limits already met in the Limits chapter. The left-hand limit $\displaystyle\lim_{x \to a^-} f(x)$ describes the value approached as $x$ increases towards $a$ from below; the right-hand limit $\displaystyle\lim_{x \to a^+} f(x)$ describes the value approached as $x$ decreases towards $a$ from above. The ordinary (two-sided) limit exists only when these two are equal:

\lim_{x \to a} f(x) = L \quad\Longleftrightarrow\quad \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L.

This is why, for a function defined by different rules on either side of $a$ (a piecewise function), continuity is tested by computing the left-hand limit, the right-hand limit and $f(a)$ separately, then checking that all three are equal.

Illustration. Consider $f(x) = x^2 + 1$ at $x = 2$ . Here $f(2) = 2^2 + 1 = 5$ (condition 1 holds); $\displaystyle\lim_{x \to 2} (x^2+1) = 5$ (condition 2 holds, the limit exists); and the two agree, $5 = 5$ (condition 3 holds). All three conditions are satisfied, so $f$ is continuous at $x = 2$ .

Note

A Limit Existing Is Not Enough — It Must Also Equal $f(a)$

A function can have a perfectly good limit at a point and still be discontinuous there, if that limit does not match the actual value $f(a)$ . Checking only that the limit exists (condition 2) and skipping the comparison with $f(a)$ (condition 3) is the single most common source of wrong answers in this chapter. Always finish the test by comparing the limit with $f(a)$ .

Maharashtra's Std XI commerce Mathematics and Statistics syllabus draws on the same continuity principles — the three-condition test, one-sided limits, and the standard-function results below — that underpin calculus across Indian higher-secondary mathematics-and-statistics curricula, scoped here to algebraic, modulus, rational and simple standard functions rather than a full analysis course.

Definition 1Continuity at a point

A function $f$ is continuous at $x=a$ when three conditions hold together: $f(a)$ is defined, $\displaystyle\lim_{x\to a} f(x)$ exists, and $\displaystyle\lim_{x\to a} f(x) = f(a)$ . Failure of any one makes $f$ discontinuous at $a$ .

Definition 2Left-hand and right-hand limits

$\displaystyle\lim_{x\to a^-} f(x)$ is the value approached from below $a$ ; $\displaystyle\lim_{x\to a^+} f(x)$ is the value approached from above $a$ . The two-sided limit exists only when both are finite and equal.

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