query: Show that if $f(x)$ is integrable in $[a,b]$, then $f(x)$ is continuous at at least one point in $[a,b]$.
positive_passages.docid: 152784
positive_passages.text: $f \in {\mathscr R[a,b]} \implies f $ has infinitely many points of continuity.
negative_passages.docid: 69869
negative_passages.text: if $f$ is continuous on $\mathbb{R}$ and $f(r)=0,r \in \mathbb{Q}$, then $f(x)=0,x \in \mathbb{R}$
negative_passages.docid: 108060
negative_passages.text: Suppose $f:[0,\infty]\rightarrow$ to R and has continuous $f'(x)$ and $f''(x)$ f(x) → 0 as x → ∞. Show that f′(x)→0 as x→∞
negative_passages.docid: 124444
negative_passages.text: If $f$ s discontinuous at $x_0$ but $g$ is continuous there, then $f+g$?
negative_passages.docid: 74364
negative_passages.text: Show that function $f$ has a continuous extension to $[a,b]$ iff $f$ is uniformly continuous on $(a,b)$
negative_passages.docid: 118332
negative_passages.text: proof $f(\overline{A})\subseteq \overline{f(A)} \Leftrightarrow f$ continuous
negative_passages.docid: 22061
negative_passages.text: $|f(x)-f(y)|\geq k|x-y|$.Then $f$ is bijective and its inverse is continuous.
negative_passages.docid: 66391
negative_passages.text: Show that $f(x+y)=f(x)+f(y)$ implies $f$ continuous $\Leftrightarrow$ $f$ measurable
negative_passages.docid: 55368
negative_passages.text: $f$ is continuous, if $f_n$ continuous and $f_n\to f$ uniformly
negative_passages.docid: 141223
negative_passages.text: $f(x^2) = 2f(x)$ and $f(x)$ continuous
negative_passages.docid: 158423
negative_passages.text: How to prove that if $f$ is continuous a.e., then it is measurable.
negative_passages.docid: 72847
negative_passages.text: Does $f(x)$ is continuous and $f = 0$ a.e. imply $f=0$ everywhere?
negative_passages.docid: 113503
negative_passages.text: Show that exists a not decreasing function that $f:(a,b)\rightarrow \mathbb{R}$ that is continuous only in $(a,b)\setminus D$.
negative_passages.docid: 102457
negative_passages.text: for any $A\subseteq X$, $f(\overline{A})\subseteq\overline{f(A)}$ , if and only if $f: X \to Y$ is continuous.
negative_passages.docid: 94781
negative_passages.text: Prove that $M(t)=\sup_ {a \leq x \leq t} f(x)$ given $f(x)$ is continuous on $[a,b]$
negative_passages.docid: 44144
negative_passages.text: If $f:[a,b]\to \mathbb R$ is continuous then so is $\max_{t\in [a,x]}f(t)$?
negative_passages.docid: 145368
negative_passages.text: How to show $f:Y\to X^{+}$ is continuous?