1. What Continuity at a Point Means
A function is said to be continuous at a point if its value at that point matches the value it approaches from nearby. In simple notes: the function behaves smoothly around that point without any jump, hole, or sudden change.
2. The Three Conditions for Continuity at x = a
To confirm continuity at a specific point, all three conditions below must be satisfied.
2.1. 1. The Function Value Exists
f(a) must be defined. If the function does not have a value at x = a, continuity fails immediately.
2.2. 2. The Limit Exists
The limit as x approaches a must exist — meaning the left-hand and right-hand limits match.
\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)
2.3. 3. The Limit Equals the Function Value
This is the most important condition:
\lim_{x \to a} f(x) = f(a)
The ‘approach value’ and the actual value must be the same.
3. Graphical Understanding
On a graph, the function is continuous at x = a if the curve passes smoothly through the point (a, f(a)). No holes, no jumps — just a clean, unbroken path.
4. Example of a Continuous Function at a Point
Consider:
f(x) = x^2
4.1. Check at x = 3
1. f(3) exists → f(3) = 9
2. Limit exists → \( \lim_{x \to 3} x^2 = 9 \)
3. Limit equals value → 9 = 9
So f(x) is continuous at x = 3.
5. Example of a Function Not Continuous at a Point
Let:
f(x) = \frac{x^2 - 4}{x - 2}
This simplifies to x + 2 for all x ≠ 2.
5.1. Check at x = 2
1. f(2) does not exist (division by 0)
2. Limit exists → \( \lim_{x \to 2} f(x) = 4 \)
3. Since f(2) doesn't exist, continuity fails.
The graph has a hole at x = 2 → removable discontinuity.
6. Left-Hand and Right-Hand View
For continuity at x = a, both sides must behave consistently:
\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)
If either one-sided limit differs, the point is discontinuous.
7. Why Continuity at a Point Matters
Understanding continuity at a point helps in analysing smooth behaviour of functions, especially before studying derivatives. It forms the base for understanding more advanced ideas in calculus.