query: Proof that a monotone function on $(-\infty,\infty)$ has at most countably many discontinuity points
positive_passages.docid: 28191
positive_passages.text: How to show that a set of discontinuous points of an increasing function is at most countable
negative_passages.docid: 13732
negative_passages.text: $||f||_p\to ||f||_\infty$ under general assumptions
negative_passages.docid: 69066
negative_passages.text: $\epsilon{\rm -}\delta$ proof of the discontinuity of Dirichlet's function
negative_passages.docid: 108997
negative_passages.text: Continuous functions that attain local extrema at every point
negative_passages.docid: 128975
negative_passages.text: Behavior of $\|f\|_p$ when $p$ is near the endpoint of $I$
negative_passages.docid: 40297
negative_passages.text: Proof of uniform limit of Continuous Functions
negative_passages.docid: 43001
negative_passages.text: Proof for Dirichlet Function and discontinuous
negative_passages.docid: 17779
negative_passages.text: Proving the Limit of a Sequence of without the Dominated Convergence Theorem
negative_passages.docid: 75882
negative_passages.text: Countability of local maxima on continuous real-valued functions
negative_passages.docid: 154572
negative_passages.text: Continuity of a function at an isolated point
negative_passages.docid: 152784
negative_passages.text: $f \in {\mathscr R[a,b]} \implies f $ has infinitely many points of continuity.
negative_passages.docid: 65033
negative_passages.text: Prove inverse of strictly monotone increasing function is continuous over the range of original function
negative_passages.docid: 120064
negative_passages.text: Proving that without the axiom of choice there is a set with an accumulation point that isn'tthe limit of a convergeant sequence
negative_passages.docid: 86698
negative_passages.text: Every uniformly continuous real function has at most linear growth at infinity
negative_passages.docid: 11167
negative_passages.text: A counterexample of a differentiable function on an interval whose derivative is discontinuous
negative_passages.docid: 57282
negative_passages.text: A Riemann integrable function must have infinitely many points of continuity
negative_passages.docid: 132824
negative_passages.text: Does $f$ monotone and $f\in L_{1}([a,\infty))$ imply $\lim_{t\to\infty} t f(t)=0$?