query: Prove a real-valued function is monotonic if it is continuous on an open interval and has no local extremes
positive_passages.docid: 18082
positive_passages.text: A continuous function that attains neither its minimum nor its maximum at any open interval is monotone
negative_passages.docid: 126212
negative_passages.text: Proving uniform continuity on an interval
negative_passages.docid: 53738
negative_passages.text: Monotone functions and continuity
negative_passages.docid: 97104
negative_passages.text: Proof that continuous function on $ [0, 2]$ has $f(x) = f(x-1)$ for some $ x$
negative_passages.docid: 104257
negative_passages.text: $C^1$ functions on an open connected set and local Lipschitz continuity
negative_passages.docid: 62118
negative_passages.text: Prove $\epsilon$-$\delta$ definition of continuity implies the open set definition for real function
negative_passages.docid: 6186
negative_passages.text: An Integrable function which is not essentially bounded on any subinterval of $[0,1]$
negative_passages.docid: 56127
negative_passages.text: Prove a function is constant under certain conditions
negative_passages.docid: 113122
negative_passages.text: Prove that a continuous real function with finite limits is bounded
negative_passages.docid: 154572
negative_passages.text: Continuity of a function at an isolated point
negative_passages.docid: 77944
negative_passages.text: Injective functions with intermediate-value property are continuous. Better proof?
negative_passages.docid: 44607
negative_passages.text: Prove that a continuous function on a closed interval attains a maximum
negative_passages.docid: 141618
negative_passages.text: Is every real valued function on an interval a sum of two functions with Intermediate Value Property?
negative_passages.docid: 58816
negative_passages.text: Existence of a continuous function which does not achieve a maximum.
negative_passages.docid: 75735
negative_passages.text: Let $f$ be a continuous function on an interval around $0$ and let $a_i=f(\frac{1}{i})$ (for large enough $i$)
negative_passages.docid: 158567
negative_passages.text: A uniform continuous function which is not Hölder continuous
negative_passages.docid: 151573
negative_passages.text: Show f is continuous on (0, ∞) iff f is continuous at 1 for a defined function.