1. What Continuity Really Means
A function is said to be continuous at a point if its graph has no break, jump, or hole around that point. When moving along the curve near that point, everything feels smooth and connected.
In personal notes: continuity means the function behaves nicely without sudden surprises.
2. Limit-Based Definition
The formal idea of continuity comes from limits. A function is continuous at a point if the value it approaches from nearby matches the actual value at that point.
2.1. Definition Using Limits
f \text{ is continuous at } x = a \text{ if } \lim_{x \to a} f(x) = f(a).
This means the function's limit and its value match perfectly.
3. Three Conditions for Continuity at a Point
To confirm that a function is continuous at x = a, all three of these must be true:
3.1. 1. The function value exists
f(a) must be defined.
3.2. 2. The limit exists
\( \lim_{x \to a} f(x) \) must exist.
3.3. 3. The limit equals the value
The limit and the actual value at that point must be the same.
\lim_{x \to a} f(x) = f(a)
4. Graphical Understanding
On a graph, continuity at a point means you can draw the curve through that point without lifting your pencil. A tiny zoom around the point shows a smooth, unbroken curve.
5. Examples to Understand Continuity
These examples show how the definition works in simple situations.
5.1. Example 1 — A Continuous Function
Let:
f(x) = x^2
At any point a:
\lim_{x \to a} x^2 = a^2 = f(a)
So the function is continuous everywhere.
5.2. Example 2 — A Function with a Hole
Let:
f(x) = \frac{x^2 - 4}{x - 2}
This simplifies to x + 2 for x ≠ 2. But at x = 2, the original expression is undefined.
So f(2) does not exist, even though the limit as x → 2 is 4. The function is not continuous at x = 2 because the first condition fails.
6. Why This Definition Is Important
Continuity ensures smooth behaviour of functions. The limit-based definition becomes the foundation for defining derivatives and understanding more advanced ideas in calculus.