Intuition of Uniform Continuity

Question

I understand the mathematical definition of Uniform Continuity. But I was wondering that if there is a intuitive explanation for this concept similar to Continuity where we sometimes say that if I draw the graph of a continuous function I do not need to lift the chalk from the board.

Answer 1

Steepness. Imagine if there was no limit to how steep a function got over an invterval. Then the function would not be uniformly continuous.

Remember if a function is continuous on a closed and bounded interval then it is uniformly continuous. This is true because this doesn't allow for asymptotes larger and larger steepness.

Examples of not uniformly continuous continuous functions:

-a function with an asymptote like $1/x$ on $(0,\infty)$.

-a function that oscillates but the oscillations get more and more frequent around $x=0$ like $\sin(1/x)$ on the interval $(0,\infty)$

-Any polynomial of degree greater than 1: $x^2$ on $\mathbb{R}$. This gets steeper and steeper as $x$ gets larger and larger.

Hope this helps.