query: Prove that $f$ is continuous on $X$ iff $int(f(A)) \subset f(intA)$.
positive_passages.docid: 142459
positive_passages.text: Continuity of a Function in Terms of Closure and Interior
negative_passages.docid: 16212
negative_passages.text: If $f$ is continuous, nonnegative on $[a, b]$, show that $\int_{a}^{b} f(x) d(x) = 0$ iff $f(x) = 0$
negative_passages.docid: 2093
negative_passages.text: If $f$ is continuous then $G$ is homeomorphic to $X$.
negative_passages.docid: 80643
negative_passages.text: Show that f(x) defined on [0,infinity) R to is continuous at 0.
negative_passages.docid: 55238
negative_passages.text: Show that if $f:X\to Y$ is a continuous function if and only if the graph of $f$ is a closed subset of $X\times Y$
negative_passages.docid: 52922
negative_passages.text: let $f \colon [a,b] \to \mathbb{R}$ be continuous on $[a,b]$ and..
negative_passages.docid: 118207
negative_passages.text: Prove that the product of two continuous functions is continuous
negative_passages.docid: 136412
negative_passages.text: Differentiable at $x=a$ implies continuous at $x=a$
negative_passages.docid: 21301
negative_passages.text: If nonnegative $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$, then $\int_0^1 \Big| \frac{f''(x)}{f(x)} \Big| \,dx >4$
negative_passages.docid: 11158
negative_passages.text: Proof that a continuous function with $f(x) = f(x^2)$ is constant.
negative_passages.docid: 62270
negative_passages.text: Given two increasing continuous functions $f,g$ prove that $(b-a) \int^b_a f(x)g(x) dx > \int^b_a f(x) dx \int^b_a g(x) dx$
negative_passages.docid: 134752
negative_passages.text: Proof (epsilon delta) for the continuity of a function at a point
negative_passages.docid: 94781
negative_passages.text: Prove that $M(t)=\sup_ {a \leq x \leq t} f(x)$ given $f(x)$ is continuous on $[a,b]$
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: 153730
negative_passages.text: Show that $f(x)=x^2$ is continuous at $a=2$ using the $\delta-\epsilon$ definition of continuity.
negative_passages.docid: 145919
negative_passages.text: If $f: \Bbb{R}\rightarrow\Bbb{R}$ is continuous at $0$ and $f(x)=f(2x)$ for each $x\in\Bbb{R}$ then $f$ is constant.
negative_passages.docid: 150235
negative_passages.text: $\epsilon$-$\delta$ proof that $f(x) = x \sin(1/x)$, $x \ne 0$, is continuous