Question

Prove that if $x,y\in X$ and $r\in\mathbb R$ are such that $f(x)<r<f(y), $ then $\exists z\in X:f(z)=r$

score 491 · Answer 1 · 2026-05-10 13:50:55

491

Views

Prove that if $x,y\in X$ and $r\in\mathbb R$ are such that $f(x)<r<f(y), $ then $\exists z\in X:f(z)=r$

Published on 10 May 2026 - 13:50

score 1.3k · Answer 2 · 2026-04-13 21:48:02

1.3k

Views

Let $f:\mathbb{R}\to\mathbb{R}$, $f(kx)=f(x)$. Show that if $f$ is continuous at $0$ then $f$ is a Constant function

Published on 13 Apr 2026 - 21:48

score 108 · Answer 3 · 2026-04-17 15:58:32

108

Views

Prove $f(x) =0$ for all $x$

Published on 17 Apr 2026 - 15:58

score 57 · Answer 4 · 2026-05-10 15:26:10

57

Views

Let $f:\Xi\rightarrow\mathbb{R}$ we consider $F$ defined by $F(\nu):=\int_{\Xi}f(\xi)\nu(d\xi)$. Is $F$ continuous with respect to the metric $W_{p}$?

Published on 10 May 2026 - 15:26

score 113 · Answer 5 · 2026-04-15 07:53:43

113

Views

The Hölder condition with the negative exponent $|f(x) - f(y)| \leq \frac{1}{|x-y|}$

Published on 15 Apr 2026 - 7:53

score 179 · Answer 6 · 2026-05-11 02:28:26

179

Views

Still struggling with continuity

Published on 11 May 2026 - 2:28

score 161 · Answer 7 · 2026-05-10 21:57:28

161

Views

Find a monotonously increasing function in the interval $[0,1]$ that is discontinuous at every point$ (x=1/n), n=[2,3...\infty]$

Published on 10 May 2026 - 21:57

score 81 · Answer 8 · 2026-05-12 23:02:50

81

Views

Defining equicontinuity in terms of sequences

Published on 12 May 2026 - 23:02

score 125 · Answer 9 · 2026-05-14 00:23:29

125

Views

Coloring $\mathbb{R}^2$ and single-colored paths

Published on 14 May 2026 - 0:23

score 38 · Answer 10 · 2026-04-16 15:15:26

38

Views

proving piece wise function is discontinuous

Published on 16 Apr 2026 - 15:15

score 92 · Answer 11 · 2026-04-10 05:47:19

92

Views

Continuity of analogous Thomae's function

Published on 10 Apr 2026 - 5:47

score 259 · Answer 12 · 2026-05-11 16:10:02

259

Views

Discontinuities in heat equation solution

Published on 11 May 2026 - 16:10

score 33 · Answer 13 · 2026-04-15 07:47:43

33

Views

Continuity of a function $f$ and $f^{-1}$ from $\mathbb{R}$ to $S^1$.

Published on 15 Apr 2026 - 7:47

score 37 · Answer 14 · 2026-04-16 01:07:35

37

Views

$a, b$ so $f(x)=x^2\sin(1/x+1), x>0$ $f(x)= ax+b, x \le 0$ becomes continuous at $x=0$

Published on 16 Apr 2026 - 1:07

score 87 · Answer 15 · 2026-04-11 02:53:01

87

Views

Showing that $f(x)$ is well-defined and continuous on $\mathbb{R} \setminus \mathbb{Z}$.

Published on 11 Apr 2026 - 2:53

List Question

Prove that if $x,y\in X$ and $r\in\mathbb R$ are such that $f(x)<r<f(y), $ then $\exists z\in X:f(z)=r$

Let $f:\mathbb{R}\to\mathbb{R}$, $f(kx)=f(x)$. Show that if $f$ is continuous at $0$ then $f$ is a Constant function

Prove $f(x) =0$ for all $x$

Let $f:\Xi\rightarrow\mathbb{R}$ we consider $F$ defined by $F(\nu):=\int_{\Xi}f(\xi)\nu(d\xi)$. Is $F$ continuous with respect to the metric $W_{p}$?

The Hölder condition with the negative exponent $|f(x) - f(y)| \leq \frac{1}{|x-y|}$

Still struggling with continuity

Find a monotonously increasing function in the interval $[0,1]$ that is discontinuous at every point$ (x=1/n), n=[2,3...\infty]$

Defining equicontinuity in terms of sequences

Coloring $\mathbb{R}^2$ and single-colored paths

proving piece wise function is discontinuous

Continuity of analogous Thomae's function

Discontinuities in heat equation solution

Continuity of a function $f$ and $f^{-1}$ from $\mathbb{R}$ to $S^1$.

$a, b$ so $f(x)=x^2\sin(1/x+1), x>0$ $f(x)= ax+b, x \le 0$ becomes continuous at $x=0$

Showing that $f(x)$ is well-defined and continuous on $\mathbb{R} \setminus \mathbb{Z}$.

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