Question

$f(0)=0$ ; $f(x):=\int_0^x \cos \frac 1t \cos \frac 3t \cos \frac 5t \cos \frac 7t\,dt$, $\forall x \ne 0$. Is $f$ differentiable at $0$?

score 125 · Answer 1 · 2016-12-09 15:52:11

125

Views

$f(0)=0$ ; $f(x):=\int_0^x \cos \frac 1t \cos \frac 3t \cos \frac 5t \cos \frac 7t\,dt$, $\forall x \ne 0$. Is $f$ differentiable at $0$?

Published on 09 Dec 2016 - 15:52

score 3.8k · Answer 2 · 2016-12-12 02:00:19

3.8k

Views

If $f,g$ are bounded and Riemann integrable, then $\max{f,g}$ and $\min{f,g}$ are Riemann integrable

Published on 12 Dec 2016 - 2:00

score 69 · Answer 3 · 2016-12-12 02:27:42

69

Views

If $f(x^2)$ is $\mathscr{R} [0,a]$, then prove $f(x^2)$ and $xf(x^2)$ are $\mathscr{R}[-a,a]$.

Published on 12 Dec 2016 - 2:27

score 124 · Answer 4 · 2016-12-12 21:14:50

124

Views

Looking for weird counterexample for Riemann integrability question

Published on 12 Dec 2016 - 21:14

score 1k · Answer 5 · 2026-03-28 06:40:09

1k

Views

Every monotonic function on $[a,b]$ is integrable?

Published on 28 Mar 2026 - 6:40

score 45 · Answer 6 · 2016-12-13 12:15:25

45

Views

Is $h(x)$ $:=$ $-1^{\lfloor x \rfloor} \over {\lfloor x+1 \rfloor}$ Riemann- and Lebesgue-integrable?

Published on 13 Dec 2016 - 12:15

score 71 · Answer 7 · 2026-03-28 06:48:15

71

Views

Analysis, difficulty proving $f(x)$ is riemann integrable on $[a,b]$

Published on 28 Mar 2026 - 6:48

score 1k · Answer 8 · 2016-12-15 18:23:40

1k

Views

Some basic properties of Riemann integrable functions

Published on 15 Dec 2016 - 18:23

score 885 · Answer 9 · 2016-12-16 06:29:38

885

Views

Example of Riemann integrable function which their composition is not Riemann integrable

Published on 16 Dec 2016 - 6:29

score 488 · Answer 10 · 2016-12-17 23:55:53

488

Views

Limit of integral: limit variable in the boundary

Published on 17 Dec 2016 - 23:55

score 260 · Answer 11 · 2016-12-18 05:10:32

260

Views

Does the limit $\lim _{x\to 0} \frac 1x \int_0^x \left|\cos \frac 1t \right| dt$ exists?

Published on 18 Dec 2016 - 5:10

score 470 · Answer 12 · 2016-12-18 20:38:22

470

Views

Riemann integral of piecewise function nonzero only at $1/n$ on $[0,1]$

Published on 18 Dec 2016 - 20:38

score 154 · Answer 13 · 2016-12-19 00:13:06

154

Views

Dominated Convergence Theorem - variable in set over which we integrate

Published on 19 Dec 2016 - 0:13

score 526 · Answer 14 · 2016-12-19 03:27:18

526

Views

If $f$ is bounded and if $f^2$ is Riemann integrable, then is f Riemann integrable?

Published on 19 Dec 2016 - 3:27

score 323 · Answer 15 · 2016-12-20 22:40:52

323

Views

Proving that if $f\in\mathscr{R} [a,b]$ then $\sqrt{f}\in \mathscr{R}[a,b]$

Published on 20 Dec 2016 - 22:40

List Question

$f(0)=0$ ; $f(x):=\int_0^x \cos \frac 1t \cos \frac 3t \cos \frac 5t \cos \frac 7t\,dt$, $\forall x \ne 0$. Is $f$ differentiable at $0$?

If $f,g$ are bounded and Riemann integrable, then $\max{f,g}$ and $\min{f,g}$ are Riemann integrable

If $f(x^2)$ is $\mathscr{R} [0,a]$, then prove $f(x^2)$ and $xf(x^2)$ are $\mathscr{R}[-a,a]$.

Looking for weird counterexample for Riemann integrability question

Every monotonic function on $[a,b]$ is integrable?

Is $h(x)$ $:=$ $-1^{\lfloor x \rfloor} \over {\lfloor x+1 \rfloor}$ Riemann- and Lebesgue-integrable?

Analysis, difficulty proving $f(x)$ is riemann integrable on $[a,b]$

Some basic properties of Riemann integrable functions

Example of Riemann integrable function which their composition is not Riemann integrable

Limit of integral: limit variable in the boundary

Does the limit $\lim _{x\to 0} \frac 1x \int_0^x \left|\cos \frac 1t \right| dt$ exists?

Riemann integral of piecewise function nonzero only at $1/n$ on $[0,1]$

Dominated Convergence Theorem - variable in set over which we integrate

If $f$ is bounded and if $f^2$ is Riemann integrable, then is f Riemann integrable?

Proving that if $f\in\mathscr{R} [a,b]$ then $\sqrt{f}\in \mathscr{R}[a,b]$

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