๐ What is Continuity at a Point?
A function, $f(x)$, is continuous at a point $x = c$ if it satisfies three key conditions:
- ๐ Condition 1: $f(c)$ is defined. In other words, the function has a value at $x = c$.
- ๐ Condition 2: $\lim_{x \to c} f(x)$ exists. This means the limit of the function as $x$ approaches $c$ exists.
- ๐ค Condition 3: $\lim_{x \to c} f(x) = f(c)$. The limit of the function as $x$ approaches $c$ must equal the function's value at $x = c$.
๐ What is Continuity on an Interval?
A function, $f(x)$, is continuous on an interval (a, b) if it is continuous at every point within that interval. For a closed interval [a, b], the function must also be continuous from the right at 'a' ($\lim_{x \to a^+} f(x) = f(a)$) and continuous from the left at 'b' ($\lim_{x \to b^-} f(x) = f(b)$).
๐ Continuity at a Point vs. Continuity on an Interval: The Breakdown
| Feature |
Continuity at a Point |
Continuity on an Interval |
| Definition |
Function is continuous at a specific $x$ value. |
Function is continuous at every $x$ value within a specified interval. |
| Conditions |
$f(c)$ is defined, $\lim_{x \to c} f(x)$ exists, and $\lim_{x \to c} f(x) = f(c)$. |
Must be continuous at every point in the interval. For closed intervals, continuity from the right at the left endpoint and continuity from the left at the right endpoint are also required. |
| Scope |
Local property (specific to a single point). |
Global property (applies across a range of points). |
| Implication |
If a function is continuous at a point, it doesn't necessarily mean it's continuous anywhere else. |
If a function is continuous on an interval, it is, by definition, continuous at every point within that interval. |
| Example |
$f(x) = \frac{x^2 - 4}{x - 2}$ is continuous at $x = 3$. |
$f(x) = x^2$ is continuous on the interval $(- \infty, \infty)$. |
๐ Key Takeaways
- ๐ Local vs. Global: Continuity at a point is a local property, while continuity on an interval is a global property.
- โ
Interval Implies Point: If a function is continuous on an interval, it *must* be continuous at every point within that interval. The reverse is not always true.
- ๐ง Endpoints Matter: When dealing with closed intervals, remember to check continuity from the right and left at the respective endpoints.