Question

Why does $|x-x_0|<\delta \Rightarrow |f(x)-f(x_0)|<\epsilon$ imply $|f(x)|>|f(x_0)|-\epsilon$ for continuous f?

score 54 · Answer 1 · 2018-05-24 15:27:58

54

Views

Why does $|x-x_0|<\delta \Rightarrow |f(x)-f(x_0)|<\epsilon$ imply $|f(x)|>|f(x_0)|-\epsilon$ for continuous f?

Published on 24 May 2018 - 15:27

score 586 · Answer 2 · 2026-03-26 04:52:17

586

Views

Uniform continuity of series

Published on 26 Mar 2026 - 4:52

score 76 · Answer 3 · 2018-05-24 17:46:54

Using modulus to answer: "How close to $\pi$ do we need $x$ to be for $x^2$ to be within $10^{−100}$ of $\pi ^2$?"

score 1k · Answer 4 · 2026-03-25 05:05:34

1k

Views

Continuity on a set is different from Pointwise Continuity on a set?

Published on 25 Mar 2026 - 5:05

score 78 · Answer 5 · 2026-03-24 22:17:33

78

Views

Under which circumstances is the space of Hölder continuous functions a Banach space?

Published on 24 Mar 2026 - 22:17

score 69 · Answer 6 · 2018-05-25 12:56:07

69

Views

the function is continuous on which set

Published on 25 May 2018 - 12:56

score 104 · Answer 7 · 2026-03-24 22:16:05

Why are in the definition of the Hölder space $C^{k+\gamma}$ only the partial derivatives of order $k$ assumed to be $\gamma$-Hölder continuous?

score 22 · Answer 8 · 2018-05-25 17:23:47

22

Views

$h$ is continuous and integral double

Published on 25 May 2018 - 17:23

score 196 · Answer 9 · 2026-03-26 04:49:35

196

Views

Uniform continuity for a function continuous only for integers

Published on 26 Mar 2026 - 4:49

score 386 · Answer 10 · 2018-05-26 07:21:16

386

Views

Why continuity of $X$ needed for $\int_{g^{-1}(y)}^\infty f_X(x) \, dx = 1-F_X(g^{-1}(y))$?

Published on 26 May 2018 - 7:21

score 87 · Answer 11 · 2026-03-26 06:27:52

87

Views

Definition of continuity on an open interval (a, b)

Published on 26 Mar 2026 - 6:27

score 70 · Answer 12 · 2026-04-01 04:37:15

70

Views

Explanation of the continuity with respect to norms

Published on 01 Apr 2026 - 4:37

score 57 · Answer 13 · 2018-05-27 09:22:34

57

Views

Dense set and continuous functions

Published on 27 May 2018 - 9:22

score 61 · Answer 14 · 2026-03-26 04:53:03

61

Views

Example of uniform continuous unbounded function such $\lim_{x \to \infty}f(x) \neq \infty$ or $\lim_{x \to \infty}f(x) \neq -\infty$

Published on 26 Mar 2026 - 4:53

score 3.2k · Answer 15 · 2026-03-25 11:17:39

3.2k

Views

Cubic root epsilon delta proof

Published on 25 Mar 2026 - 11:17

List Question

Why does $|x-x_0|<\delta \Rightarrow |f(x)-f(x_0)|<\epsilon$ imply $|f(x)|>|f(x_0)|-\epsilon$ for continuous f?

Uniform continuity of series

Using modulus to answer: "How close to $\pi$ do we need $x$ to be for $x^2$ to be within $10^{−100}$ of $\pi ^2$?"

Continuity on a set is different from Pointwise Continuity on a set?

Under which circumstances is the space of Hölder continuous functions a Banach space?

the function is continuous on which set

Why are in the definition of the Hölder space $C^{k+\gamma}$ only the partial derivatives of order $k$ assumed to be $\gamma$-Hölder continuous?

$h$ is continuous and integral double

Uniform continuity for a function continuous only for integers

Why continuity of $X$ needed for $\int_{g^{-1}(y)}^\infty f_X(x) \, dx = 1-F_X(g^{-1}(y))$?

Definition of continuity on an open interval (a, b)

Explanation of the continuity with respect to norms

Dense set and continuous functions

Example of uniform continuous unbounded function such $\lim_{x \to \infty}f(x) \neq \infty$ or $\lim_{x \to \infty}f(x) \neq -\infty$

Cubic root epsilon delta proof

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