Questions tagged [continuity]

Intuitively, a continuous function is one where small changes of input result in correspondingly small changes of output. Use this tag for questions involving this concept. As there are many mathematical formalizations of continuity, please also use an appropriate subject tag such as (real-analysis) or (general-topology)

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Is function $f(x) = \frac{x-1}{x-1}$ defined at $x=1$? [duplicate]

I think the answer should be YES. Because the function is actually $f(x) = 1$ Likewise, the function $f(x) = \frac{x^2-1}{x-1}$ is identical to $f(x) = x+1$. Therefore it is well defined at $x=1$. Can ...
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Bump functions of maximal height under smoothness constraints

Suppose I want to find a function that is of maximal height at $0$ and of zero height outside of the unit ball. In other words, a bump function of maximal height. The constraint I put on this function ...
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Prove/disprove: $d(x, \phi(x)) \leq f(x) - f(\phi(x))$ implies $\phi$ is a contraction

According to Approach0, this question seems new. There is a similar question, however in that question asked for a proof about a function, which satisfied the definition (see Problem), has a fix point....
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Suppose that $f(x)$ is continuous in $[0, 1]$ and $f(0)=0=f(1)$. Prove $f(c)=1−2c^2$ for some $c\in(0,1)$ [closed]

Please someone tell me how to solve this question and most importantly the reasons behind steps i tried to solve it by making another function g(x) = f(x) - (1-2x^2) but i don't know after this step ...
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Is a continuous real function with vanishing derivative in all but countably many points constant?

The Cantor function is a standard example of a function $f:[0,1]\to\mathbb{R}$ that is continuous, has almost everywhere zero derivative, and nonetheless is not constant. More specifically, the number ...
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continuous function on $\mathbb{R}^*$

I am having a confusion, what I know is that the function $f(x) = \dfrac{1}{x}$ is continuous on $\mathbb{R^*}$, since $\mathbb{R^*}$ is its domain. But when I was discussing this with one of my ...
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Show that if a subset of $C([0,1])$ is open relative to one norm it is also open relative to another or vice versa

I am struggling coming up with a solution to the following question. Consider the two norms $||f||_1 = \int_0^1 |f(x)| \mathrm{d}x$ and $||f||_\infty = \sup_{x \in [0, 1]} |f(x)|$ on $C([0,1])$ (the ...
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Proving differentiability at a point - Confusion? [duplicate]

(For a more fleshed-out example demonstrating my confusion, see this post. In John's answer, why do we have the "smoothness" condition? Wouldn't this be necessary if and only when dealing ...
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