# Tagged Questions

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### Does bounded variation and continuous means total variation continuous

$F$ is of bounded variation and continuous. Is it true that total variation is continuous ? In case, $F$ is absolutely continuous it is trivial to see. But for the above case how to proceed ?
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### Continuity and Inverse Preservation of Boundedness Implies Preservation of Closed Sets

Sorry about the rather long title - wasn't sure what else to call it! Here is my question: Let $f : \Bbb R^n \rightarrow \Bbb R^m$ be continuous and such that $f^{-1}(F)$ is bounded whenever $F$ ...
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### Does extreme value theorem hold when continuous is replaced with bounded?

The extreme value theorem says that if the domain of a 'continuous' function is compact then both the max and min of the function lies in the domain set. My question is: can the 'continuity' be ...
First, the definitions: $f$ is AC on $E$ if $$\forall \epsilon >0\ \exists \delta >0\ \forall \{[a_k,b_k]\}_{k=1}^N \mbox{ such that }a_k,b_k \in E,\ \Sigma(b_k - a_k) <\delta : \Sigma| ... 1answer 290 views ### Uniform limit of continuous functions bounded variation Prove or disprove that if f:[a,b]\rightarrow\mathbb{R} is the uniform limit of a sequence of continuous functions each of which is of bounded variation, then f is of bounded variation on [a,b]. 2answers 279 views ### Continuously differentiable functions of bounded variation From this question, we know that a continuous function of bounded variation is not necessarily absolutely continuous. But the example (Devil's staircase) given is not differentiable. What if we ... 1answer 757 views ### proving f is absolutely continuous on [0,1] I don't seem to find version of this problem in the site, but I am sure this is pretty standard type of question. f be of bounded variation on [0,1], and f is absolutely continuous (AC) on ... 1answer 184 views ### Let f be a real valued sequentially continuous function relative to a closed bounded interval I=[a,b]. Prove that the set f(I) is bounded above The hint that I've been given is: for each n in the naturals, use the assumption that n is not an upper bound for f(I) to choose a sequence of x_n (from n=1 to infinity) in I; then apply ... 2answers 163 views ### f(0)=0,\;\;f(x)=e^{-2/x}\sin\left(e^{1/x}\right), is f bounded variation on [0,1]? Let$$f(0)=0,\;\;f(x)=e^{-2/x}\sin\left(e^{1/x}\right), is $f$ bounded variation on $[0,1]$? Here is my thinking: Since $f$ is differentiable on $(0,1]$ and continuous on $[0,1]$ If $f^\prime$ ...
I have been doing some excercises on total variation when the following questions came up to my mind: (1) Let $f$ be continuous on the interval $[0,1]$ and be of bounded variation. Is it true that ...