query: Prove that if $f$ is a bounded function on $[a,b]$, and for each $c \in (a,b)$, $f \in R[a,c]$, then $f \in R[a,b]$.
positive_passages.docid: 144322
positive_passages.text: Prove for bounded $f$, if $f:[a,r] \to \mathbb{R}$ is Riemann-Integrable for $r \in [a,b)$ then $f:[a.b] \to \mathbb{R}$ is Riemann-Integrable
negative_passages.docid: 111756
negative_passages.text: How do I prove that then $f(x) = 1$ for some $x ∈ [0, 1]$?
negative_passages.docid: 76148
negative_passages.text: Show that $\exists c \in \mathbb{R} : f(c) = 0$
negative_passages.docid: 81974
negative_passages.text: Let $f_1 , f_2: I\mapsto \mathbb{R}$ bounded functions. Show that $L(f_1)+L(f_2)\leq L(f_1+f_2)$ (Riemann integral)
negative_passages.docid: 52711
negative_passages.text: Prove that the sequence $\{v_n\}$ converges to the maximum value of $f$ on $[a,b]$
negative_passages.docid: 98399
negative_passages.text: Suppose $f$ and $g$ are entire functions, and $|f(z)| \leq |g(z)|$ for all $z \in \mathbb{C}$, Prove that $f(z)=cg(z)$.
negative_passages.docid: 92089
negative_passages.text: If $f:[a,b]\to \mathbb{R}$ is continuous and nonnegative and $\int_a^b{f}=0$, then $f(x)=0$ for all $x\in [a,b]$
negative_passages.docid: 104914
negative_passages.text: Abount linear functional: If $T(B)$ is bounded, is $T$ bounded?
negative_passages.docid: 83311
negative_passages.text: Compactness of $A:=${$f \in C[0,1], |f|_\infty \le K, |f'|_\infty \le M$}
negative_passages.docid: 61799
negative_passages.text: Prove that $\Vert f(b)-f(a)-f'(a)(b-a) \Vert\leq \sup_{x\in [a,b]} \Vert f''(x)\Vert\Vert b-a\Vert^2$
negative_passages.docid: 28620
negative_passages.text: Function $f(z)$ such that $|f(z)| > |z|$ for every $z\in\mathbb{C}, 1/f(z)$
negative_passages.docid: 95970
negative_passages.text: Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function. If $f(0)=f(1)$ then exists $x\in [0,1]$ s.t. $f(x) = f(x+\frac{1}{2})$
negative_passages.docid: 142645
negative_passages.text: $f_n$ converges uniformly on $[a,b]$ to some function $f$
negative_passages.docid: 92376
negative_passages.text: Let $f:A \rightarrow B$ and C ⊂A. Define f[C]={b∈B: b=f(a) for some a∈C}. Prove or disprove each of the statement
negative_passages.docid: 143674
negative_passages.text: Space of bounded functions is complete
negative_passages.docid: 47135
negative_passages.text: Prove if a = b, then f(a) = f(b) for any function f (with natural deduction)
negative_passages.docid: 129073
negative_passages.text: If $f'$ is bounded, then $f$ is uniformly continuous