# Tagged Questions

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### How to indicate which is the best bound for accuracy either the max or min bounds?

A Few lists of precipitation data (P-data at different stations) in descending order is used to estimate streamflow at its corresponding stations. For every P data, 2 streamflow values are ...
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### Proof Function is Bounded/Unbounded

How can I prove that the function $$\sigma_i\left(t\right) = k_i\left[\left(a+b\left(T_i-t\right)\right)e^{-c\left(T_i-t\right)}+d\right]$$ is bounded/unbounded? Note: $\sigma_i\left(t\right)$ is ...
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### Prove absolute continuity without Banach-Zarecki

Let $f$ be a real-valued continuous function of bounded variation on $[a,b]$. Suppose $f$ is absolutely continuous on $[a+\eta,b]$ for every $\eta\in(0,b-a)$. Show that $f$ is absolutely continuous on ...
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### Construct a non-monotone continuous function of bounded variation

Construct a continuous function of bounded variation on $[0,1]$ which is not monotone in any subinterval. We can follow the pattern of the Cantor-Lebesgue function (somewhat). For example, at the ...
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### Does differentiability imply having bounded variation on some subinterval?

Suppose that $f:(a,b)\to\mathbb{R}$ is a differentiable function. Does it follow that $f$ has bounded variation on some subinterval $[c,d]\subset (a,b)$? Details and ideas Being differentiable ...
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### Understanding Lemma: $\left\lVert f \right\rVert_{\infty} \leq \left|f(a) \right| + V_{a}^{b} f$

I'm learning about functions of bounded variations and need help to understand the proof of this lemma: Lemma. If $f : [a,b] \rightarrow \mathbb{R}$ is of bounded variation, then f is also ...
### Let $f$ be of bounded variation on $[a,b]$, and define $v(x) = f_{[a,x]}$. show $\int_a^b |f'|\leq TV(f).$
Let $f$ be of bounded variation on $[a,b]$, and define $v(x) = TV(f_{[a,x]})$ for all $x \in [a,b]$. show that $|f'| \leq v'$ a.e. on $[a,b]$, and infer from this that $$\int_a^b |f'|\leq TV(f).$$ ...