Uniform continuous and ordinary continuous

• Uniform continuous
• Ordinary continuous
• Difference

Uniform continuous

Uniform continuity means: If a function $ f $ is uniformly continuous, then for any two points $ x $ and $ y $ in the domain, as long as $ x $ and $ y $ are sufficiently close, $ f(x) $ and $ f(y)$ will also be sufficiently close.

Another definition using neighborhoods:

• For any real number $ \epsilon > 0 $, there exists a real number $ \delta > 0 $ such that whenever $ |x - y| < \delta $, we have $ |f(x) - f(y)| < \epsilon $.

The Heine-Cantor theorem states:

• If a function is continuous on a closed interval, then it is also uniformly continuous.

Ordinary continuous

The intuitive meaning of ordinary continuity is:

If a function $f$ is continuous, then for any point $x$ in the domain, there exists a sufficiently small neighborhood around $x$ such that $f(x)$ is sufficiently close to the values within this neighborhood.

The definition of ordinary continuity by neighborhood:

• For any real number $\epsilon > 0$ and any point $x$ in the domain, there exists a real number $\delta > 0$ such that whenever $|x - y| < \delta$, we have $|f(x) - f(y)| < \epsilon$, where \(y\) is any point in the neighborhood of $x$.

Difference

From this, we can see:

• In the definition of uniform continuity, $\delta$ only depends on $\epsilon$.
• In the definition of ordinary continuity, $\delta$ depends on both $\epsilon$ and $x$.
• Uniform continuity is a stricter condition than ordinary continuity: a uniformly continuous function is always ordinary continuous, but a ordinary continuous function is not necessarily uniformly continuous.
• The distinction between uniform continuity and continuity is not easy to understand. However, it can be better grasped through a geometric interpretation. For example, consider the function $f(x) = 1/x$. This function is continuous on the interval $(0, 1)$, but it is not uniformly continuous because choosing two points close to 0 that are close to each other does not result in their function values being sufficiently close.