query: An increasing function on a closed interval has countable many points of discontinuity.
positive_passages.docid: 150346
positive_passages.text: If $f$ is an increasing function defined on $[a,b]$ . Then, $f$ has countable number of discontinuities
negative_passages.docid: 133231
negative_passages.text: Discontinuity of Thomae Function on $\mathbb{Q}$
negative_passages.docid: 27800
negative_passages.text: For continuous function $ f:\mathbb S^1 \to \mathbb R$ there exists uncountably many distinct points $x,y$ such that $f(x)=f(y)$
negative_passages.docid: 35133
negative_passages.text: Every increasing function from a certain set to itself has at least one fixed point
negative_passages.docid: 53738
negative_passages.text: Monotone functions and continuity
negative_passages.docid: 18222
negative_passages.text: Set of continuity points of a real function
negative_passages.docid: 9320
negative_passages.text: Extension of continuous function on a closed subset of $\mathbb{R}$
negative_passages.docid: 128482
negative_passages.text: The set of subsequential limits is closed
negative_passages.docid: 69283
negative_passages.text: The limit points of an open interval (open set)
negative_passages.docid: 118400
negative_passages.text: Is there a monotonic function discontinuous over some dense set?
negative_passages.docid: 142459
negative_passages.text: Continuity of a Function in Terms of Closure and Interior
negative_passages.docid: 14896
negative_passages.text: Prove: the set of zeros of a continuous function is closed.
negative_passages.docid: 48764
negative_passages.text: Increasing real valued function whose image set is connected
negative_passages.docid: 28413
negative_passages.text: Limit of a continuous function
negative_passages.docid: 76607
negative_passages.text: Show that exists a continuous function in $E$ that is not bounded
negative_passages.docid: 66951
negative_passages.text: Proving the existence of a sequence of functions converging to 0 with an added property
negative_passages.docid: 96729
negative_passages.text: Changing the values of an integrable function $f:[a,b] \to \mathbb R$ countably infinitely many points not a dense subset of $[a,b]$