Question

Can we use Lipschitz continuity to prove $f(x,y) = x^2+y^2$ is continuous for $x,y\in \mathbb{R}$?

score 181 · Answer 1 · 2026-03-25 00:24:26

181

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Can we use Lipschitz continuity to prove $f(x,y) = x^2+y^2$ is continuous for $x,y\in \mathbb{R}$?

Published on 25 Mar 2026 - 0:24

score 54 · Answer 2 · 2026-03-25 17:42:13

54

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The function $f$ defined on $[0,1]$ to be $t \rightarrow \sum \limits_{i=1}^{n}|tx_i+(1-t)y_i|^p$ attains its maximum value at either $0$ or $1$

Published on 25 Mar 2026 - 17:42

score 131 · Answer 3 · 2026-03-24 23:40:54

131

Views

Midpoint convex and monotonicfunction

Published on 24 Mar 2026 - 23:40

score 42 · Answer 4 · 2018-03-22 12:23:57

Behaviour of a continous function $f \text{ : } \lim_{n \to \infty} f(\frac{1}{2n}) = 1$ and $\lim_{n \to \infty} f(\frac{1}{2n+1}) = 2$

score 158 · Answer 5 · 2018-03-22 12:32:15

158

Views

Discontinuity set of an increasing right continuous function

Published on 22 Mar 2018 - 12:32

score 313 · Answer 6 · 2026-03-29 15:43:11

313

Views

$f$ has no interior extreme point, therefore, $f$ is strictly increasing or decreasing function

Published on 29 Mar 2026 - 15:43

score 347 · Answer 7 · 2026-03-27 18:26:18

347

Views

Continuous function over compact subset and non-degenerate critical points

Published on 27 Mar 2026 - 18:26

score 95 · Answer 8 · 2026-03-29 15:40:35

95

Views

Heine theorem proof

Published on 29 Mar 2026 - 15:40

score 1k · Answer 9 · 2018-03-22 20:31:54

1k

Views

Show that if the restrictions of $f$ to $S_1$ and to $S_2$ are continuous then $f$ is continuous.

Published on 22 Mar 2018 - 20:31

score 517 · Answer 10 · 2026-03-25 12:30:30

517

Views

Cantor Set and Continuity

Published on 25 Mar 2026 - 12:30

score 27 · Answer 11 · 2018-03-23 15:25:31

27

Views

Continuity of finite number of functions

Published on 23 Mar 2018 - 15:25

score 17 · Answer 12 · 2018-03-23 16:29:56

17

Views

Proving continuity and differentiability of a function at a point

Published on 23 Mar 2018 - 16:29

score 50 · Answer 13 · 2018-03-23 16:41:51

50

Views

Determining continuity in a trigonometric function

Published on 23 Mar 2018 - 16:41

score 52 · Answer 14 · 2018-03-23 19:03:12

52

Views

Constant continuous functions

Published on 23 Mar 2018 - 19:03

score 51 · Answer 15 · 2026-03-25 13:42:09

51

Views

About the continuity of a partial derivative

Published on 25 Mar 2026 - 13:42

List Question

Can we use Lipschitz continuity to prove $f(x,y) = x^2+y^2$ is continuous for $x,y\in \mathbb{R}$?

The function $f$ defined on $[0,1]$ to be $t \rightarrow \sum \limits_{i=1}^{n}|tx_i+(1-t)y_i|^p$ attains its maximum value at either $0$ or $1$

Midpoint convex and monotonicfunction

Behaviour of a continous function $f \text{ : } \lim_{n \to \infty} f(\frac{1}{2n}) = 1$ and $\lim_{n \to \infty} f(\frac{1}{2n+1}) = 2$

Discontinuity set of an increasing right continuous function

$f$ has no interior extreme point, therefore, $f$ is strictly increasing or decreasing function

Continuous function over compact subset and non-degenerate critical points

Heine theorem proof

Show that if the restrictions of $f$ to $S_1$ and to $S_2$ are continuous then $f$ is continuous.

Cantor Set and Continuity

Continuity of finite number of functions

Proving continuity and differentiability of a function at a point

Determining continuity in a trigonometric function

Constant continuous functions

About the continuity of a partial derivative

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