query: If $f$ is differntiable at $x_0$, then $f$ is continuous at $x_0$. How does this work?
positive_passages.docid: 144013
positive_passages.text: If $f$ is differentiable at $x = x_0$ then $f$ is continuous at $x = x_0$.
negative_passages.docid: 94737
negative_passages.text: How to show $f$ is continuous at $x$ IFF for any sequence ${x_n}$ in $X$ converging to $x$ the sequence $f(x_n)$ converges in $Y$ to $f(x)$
negative_passages.docid: 37850
negative_passages.text: Does the everywhere differentiability of $f$ imply it is absolutely continuous on a compact interval?
negative_passages.docid: 30965
negative_passages.text: $f$ is continuous on $X$ iff $f$ is continuous on every compact subset
negative_passages.docid: 113687
negative_passages.text: $f$ is continuous function from $ [0,1]$ to itself and f(0)=0, f(1)=1 and $f(f(x))=x$ then show that $f(x)=x$ on$[0,1]$
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: 97104
negative_passages.text: Proof that continuous function on $ [0, 2]$ has $f(x) = f(x-1)$ for some $ x$
negative_passages.docid: 105652
negative_passages.text: Suppose $f: \mathbb{R} \to \mathbb{R}$ is function such that $f^3$ is continuous on $\mathbb{R}$. Prove that $f$ is continuous on $\mathbb{R}$
negative_passages.docid: 72847
negative_passages.text: Does $f(x)$ is continuous and $f = 0$ a.e. imply $f=0$ everywhere?
negative_passages.docid: 4636
negative_passages.text: How to show that $f$ is differentiable at $x=0.$
negative_passages.docid: 8836
negative_passages.text: If $f$ is continuous on $[a , b]$ and $\int_a^b f(x) p(x)dx = 0$ then $f = 0$
negative_passages.docid: 74950
negative_passages.text: Show a function for which $f(x + y) = f(x) + f(y) $ is continuous at zero if and only if it is continuous on $\mathbb R$
negative_passages.docid: 106060
negative_passages.text: If $f$ is continuous and with a limit at infinity then $f$ is uniformly continuous
negative_passages.docid: 141223
negative_passages.text: $f(x^2) = 2f(x)$ and $f(x)$ continuous
negative_passages.docid: 14578
negative_passages.text: Proving that an additive function $f$ is continuous if it is continuous at a single point
negative_passages.docid: 129551
negative_passages.text: Does $\lim_{h\rightarrow 0}\ [f(x+h)-f(x-h)]=0$ imply that $f$ is continuous?
negative_passages.docid: 148266
negative_passages.text: Let $f:X\to X $ be continuous. Does $f $ have a fixed point when $X=[0,1)$ or $X=(0,1) $?