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There exist nondecreasing functions f such that f0 = 0 almost every-where and f(b) − f(a) > 0. Simple step functions are examples of such functions. More interesting are Cantor functions, which are nondecreasing and continuous, with f(b) − f(a) > 0 and f0(t) = 0 almost everywhere. We
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In mathematics, smooth functions (also called infinitely differentiable functions) and analytic functions are two very important types of functions. One can easily prove that any analytic function of a real argument is smooth. The converse is not true, as demonstrated with the counterexample below.
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Let f be a real-valued continuous function defined on the unit interval [0, 1]. It seems intuitively clear that f should be monotonic on some subinterval I of [0, 1]. Most of the concrete examples seem to support this. A counterexample is termed nowhere monotonic, meaning that the function is not monotonic in any subinterval of [0, 1].
then the function is not one-to-one. • If no horizontal line intersects the graph of the function more than once, then the function is one-to-one. What are One-To-One Functions? Algebraic Test Deﬁnition 1. A function f is said to be one-to-one (or injective) if f(x 1) = f(x 2) implies x 1 = x 2. Lemma 2. The function f is one-to-one if and ...
This function has a limit at the origin, it is even continuous there, but it is not differentiable there as the slopes in the appropriate limit oscillate between 0 and 1 (details are almost the same as in a similar example in the section on "saw-like" functions). Modification 2. Define This function looks like this: If $$f\left( x \right)$$ is not continuous at $$x = a$$, then $$f\left( x \right)$$ is said to be discontinuous at this point. Figures $$1 – 4$$ show the graphs of four functions, two of which are continuous at $$x =a$$ and two are not. Fig 1. Continuous function. Fig 2. Discontinuous function. Fig 3. Continuous function. Fig 4. Discontinuous ...