Here is a old post regarding this questions below
Ques: Suppose that f is a continuous real-valued function defined on the closed interval [0,1]. Which of the following must be true ?
III. There is a constant E > 0 such that |f(x) -f(y)| <= E |x-y| for all x and y in [0,1]
The answer rejected III. The discussion said that not all the function are uniformly continuous.
But if we rewrite the condition as:
|f(x) -f(y)| / |x-y| = |K| <= E
So the left hand side become the abs of the slope. I am wondering for a continuous function defined on a closed interval, is it possible that the slope in not bounded?