1.11 defining continuity at a point

Lemon

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02-02-2025

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Understanding continuity is fundamental to calculus. While the intuitive notion of continuity—a function being "unbroken"—is helpful, a rigorous mathematical definition is crucial for advanced work. This post delves into the precise definition of continuity at a point, exploring its implications and providing illustrative examples.

What Does Continuity at a Point Really Mean?

Intuitively, a function is continuous at a point if you can draw its graph without lifting your pen. However, this visual approach fails for complex functions or those defined piecewise. A more robust definition relies on limits.

A function f(x) is continuous at a point x = c if and only if the following three conditions are met:

1. f(c) is defined: The function must have a value at the point c. This seems obvious, but it's crucial. If the function is undefined at c, it cannot be continuous there.

2. lim_x→c f(x) exists: The limit of the function as x approaches c must exist. This means the left-hand limit and the right-hand limit must be equal. If the limit doesn't exist (e.g., due to a jump discontinuity or an asymptote), the function is not continuous at c.

3. lim_x→c f(x) = f(c): The limit of the function as x approaches c must equal the function's value at c. This ensures there are no "holes" or "jumps" in the graph at x = c. The value of the function at the point must match the value the function is approaching.

Illustrative Examples

Let's consider a few examples to solidify our understanding:

Example 1: A Continuous Function

Consider the function f(x) = x². Let's check for continuity at x = 2.

1. f(2) = 2² = 4 (defined)
2. lim_x→2 x² = 4 (limit exists)
3. lim_x→2 x² = f(2) = 4 (limit equals function value)

Therefore, f(x) = x² is continuous at x = 2. In fact, this function is continuous everywhere.

Example 2: A Discontinuous Function (Removable Discontinuity)

Consider the function:

f(x) = (x² - 4) / (x - 2)  if x ≠ 2
f(x) = 5                 if x = 2

Let's check continuity at x = 2.

1. f(2) = 5 (defined)
2. lim_x→2 (x² - 4) / (x - 2) = lim_x→2 (x + 2) = 4 (limit exists)
3. lim_x→2 f(x) ≠ f(2) (limit does not equal function value)

Because the limit and the function value at x=2 are different, this function is discontinuous at x=2, even though the discontinuity is removable. We could redefine f(2)=4 to make it continuous.

Example 3: A Discontinuous Function (Jump Discontinuity)

Consider the piecewise function:

f(x) = x  if x < 2
f(x) = x + 1 if x ≥ 2

At x = 2:

1. f(2) = 3 (defined)
2. lim_x→2⁻ f(x) = 2 (left-hand limit)
3. lim_x→2⁺ f(x) = 3 (right-hand limit)

Since the left-hand and right-hand limits are unequal, the limit at x = 2 does not exist. Therefore, the function is discontinuous at x = 2.

Beyond a Single Point: Continuity on an Interval

A function is considered continuous on an open interval (a, b) if it's continuous at every point within that interval. Continuity on a closed interval [a, b] requires continuity on (a, b) plus the existence of one-sided limits at the endpoints a and b that match the function values at those endpoints.

Understanding continuity at a point forms the foundation for understanding more advanced concepts in calculus, such as differentiability and the Intermediate Value Theorem. Mastering this definition is key to success in higher-level mathematics.