query: Continuity question: Show that $f(x)=0, \forall x\in\mathbb{R}$.
positive_passages.docid: 2321
positive_passages.text: Can there be two distinct, continuous functions that are equal at all rationals?
negative_passages.docid: 145368
negative_passages.text: How to show $f:Y\to X^{+}$ is continuous?
negative_passages.docid: 71225
negative_passages.text: $f(0)=0$ and $\lim_{x \to \infty}f(x)=0$. Show there's $x_0$ such that $f'(x_0)=0$
negative_passages.docid: 141223
negative_passages.text: $f(x^2) = 2f(x)$ and $f(x)$ continuous
negative_passages.docid: 111756
negative_passages.text: How do I prove that then $f(x) = 1$ for some $x ∈ [0, 1]$?
negative_passages.docid: 81223
negative_passages.text: Real Analysis Question: continuity of $f(x) = \sum_{q_n<x} 1/n^2$
negative_passages.docid: 151958
negative_passages.text: How to show $f'(0)$ exist and is equal to $1$?
negative_passages.docid: 158888
negative_passages.text: Prove that the function $f(x)=\frac{x}{x^2+1}$ is continuous at $x=-1$.
negative_passages.docid: 40209
negative_passages.text: $f : S^1 \to\mathbb R$ is continuous then $f(x)=f(-x)$ for some $x\in S^1$
negative_passages.docid: 136980
negative_passages.text: Prove there exists $a\in \Bbb{R}$ such that $f'(a)=0$.
negative_passages.docid: 7358
negative_passages.text: For each $y \in \mathbb{R}$ either no $x$ with $f(x) = y$ or two such values of $x$. Show that $f$ is discontinuous.
negative_passages.docid: 30860
negative_passages.text: Prove that $f(x)=0$ in $[-1,1]$.
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: 68165
negative_passages.text: Find a function $f: \mathbb{R} \to \mathbb{R}$ that is continuous at precisely one point?
negative_passages.docid: 55605
negative_passages.text: If $f_n(x_n) \to f(x)$ whenever $x_n \to x$, show that $f$ is continuous
negative_passages.docid: 104510
negative_passages.text: For continuous $f$, $\int_0^x f(t)dt = ∫_x^1f(t)dt$ implies $f=0$.
negative_passages.docid: 85739
negative_passages.text: Show that $f(x) = 0$ for all $x \in [a,b]$ given $|f'(x)| \leq C|f(x)| $