Mixing
Sufficient condition to show that f(x) is monotonic in some neighbourhood of x
Clash Royale CLAN TAG#URR8PPP
up vote
2
down vote
favorite
I’m actually referring to another a question from here: Sufficient condition to show $f$ is monotonically increasing in some neighborhood
I know a counterexample has already been given, but I still do not fully understand why or how it is necessary for f’(x) to be continuous. Why does it have to be continuous in the whole interval, and not just at point x?
Actually I initially thought differentiability of f(x) was sufficient. Below is my reasoning.
Let $f : left[a,bright] to Bbb R$ be differentiable at some $c in left(a,bright)$, with $f’(c) > 0$. Then,
$$lim_xto cfracf(x) - f(c)x - c = L gt 0$$
So for $epsilon = L gt 0, exists delta gt 0 :$
$$left|x - cright| lt delta Rightarrow left| fracf(x) - f(c)x - c - Lright| lt L Rightarrow 0 lt fracf(x) - f(c)x - c lt 2L$$
So when $-delta lt x - c lt 0 Rightarrow x lt c$, and as $fracf(x) - f(c)x - c gt 0 Rightarrow f(x) - f(c) lt 0 Rightarrow f(x) lt f(c)$. Similarly when $0 lt x - c lt delta Rightarrow x gt c$ and $f(x) gt f(c)$. Hence we have found a neighbourhood $left|x - cright| lt delta$ in which $f(x)$ is increasing monotonically
Can someone please point out where I went wrong?
calculus limits derivatives continuity monotone-functions
asked Jul 26 at 17:53
user63858
183
add a comment |Â
up vote
2
down vote
favorite
I’m actually referring to another a question from here: Sufficient condition to show $f$ is monotonically increasing in some neighborhood
I know a counterexample has already been given, but I still do not fully understand why or how it is necessary for f’(x) to be continuous. Why does it have to be continuous in the whole interval, and not just at point x?
Actually I initially thought differentiability of f(x) was sufficient. Below is my reasoning.
Let $f : left[a,bright] to Bbb R$ be differentiable at some $c in left(a,bright)$, with $f’(c) > 0$. Then,
$$lim_xto cfracf(x) - f(c)x - c = L gt 0$$
So for $epsilon = L gt 0, exists delta gt 0 :$
$$left|x - cright| lt delta Rightarrow left| fracf(x) - f(c)x - c - Lright| lt L Rightarrow 0 lt fracf(x) - f(c)x - c lt 2L$$
So when $-delta lt x - c lt 0 Rightarrow x lt c$, and as $fracf(x) - f(c)x - c gt 0 Rightarrow f(x) - f(c) lt 0 Rightarrow f(x) lt f(c)$. Similarly when $0 lt x - c lt delta Rightarrow x gt c$ and $f(x) gt f(c)$. Hence we have found a neighbourhood $left|x - cright| lt delta$ in which $f(x)$ is increasing monotonically
Can someone please point out where I went wrong?
calculus limits derivatives continuity monotone-functions
asked Jul 26 at 17:53
user63858
183
1
Something that jumps out to me immediately is that you haven't actually shown that $f$ is monotonic in the neighborhood. In particular, you need to show that for any $x,y in (c- delta, c + delta)$, we have $x < y Rightarrow f(x) < f(y)$. All you have done is shown that this works when comparing points in the interval to $c$.
– Joe
Jul 26 at 18:04
@Joe oh right i just realised that. So the only way to show that the function is monotonically increasing is to prove that it has a positive derivative in that interval right?
– user63858
Jul 26 at 18:11
No not necessarily. Often times it's easier to show a function is monotone directly from the definition as opposed to finding the derivative. See the Cantor Function for example.
– Joe
Jul 26 at 18:17
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I’m actually referring to another a question from here: Sufficient condition to show $f$ is monotonically increasing in some neighborhood
I know a counterexample has already been given, but I still do not fully understand why or how it is necessary for f’(x) to be continuous. Why does it have to be continuous in the whole interval, and not just at point x?
Actually I initially thought differentiability of f(x) was sufficient. Below is my reasoning.
Let $f : left[a,bright] to Bbb R$ be differentiable at some $c in left(a,bright)$, with $f’(c) > 0$. Then,
$$lim_xto cfracf(x) - f(c)x - c = L gt 0$$
So for $epsilon = L gt 0, exists delta gt 0 :$
$$left|x - cright| lt delta Rightarrow left| fracf(x) - f(c)x - c - Lright| lt L Rightarrow 0 lt fracf(x) - f(c)x - c lt 2L$$
So when $-delta lt x - c lt 0 Rightarrow x lt c$, and as $fracf(x) - f(c)x - c gt 0 Rightarrow f(x) - f(c) lt 0 Rightarrow f(x) lt f(c)$. Similarly when $0 lt x - c lt delta Rightarrow x gt c$ and $f(x) gt f(c)$. Hence we have found a neighbourhood $left|x - cright| lt delta$ in which $f(x)$ is increasing monotonically
Can someone please point out where I went wrong?
calculus limits derivatives continuity monotone-functions
asked Jul 26 at 17:53
user63858
183
I’m actually referring to another a question from here: Sufficient condition to show $f$ is monotonically increasing in some neighborhood
I know a counterexample has already been given, but I still do not fully understand why or how it is necessary for f’(x) to be continuous. Why does it have to be continuous in the whole interval, and not just at point x?
Actually I initially thought differentiability of f(x) was sufficient. Below is my reasoning.
Let $f : left[a,bright] to Bbb R$ be differentiable at some $c in left(a,bright)$, with $f’(c) > 0$. Then,
$$lim_xto cfracf(x) - f(c)x - c = L gt 0$$
So for $epsilon = L gt 0, exists delta gt 0 :$
$$left|x - cright| lt delta Rightarrow left| fracf(x) - f(c)x - c - Lright| lt L Rightarrow 0 lt fracf(x) - f(c)x - c lt 2L$$
So when $-delta lt x - c lt 0 Rightarrow x lt c$, and as $fracf(x) - f(c)x - c gt 0 Rightarrow f(x) - f(c) lt 0 Rightarrow f(x) lt f(c)$. Similarly when $0 lt x - c lt delta Rightarrow x gt c$ and $f(x) gt f(c)$. Hence we have found a neighbourhood $left|x - cright| lt delta$ in which $f(x)$ is increasing monotonically
Can someone please point out where I went wrong?
calculus limits derivatives continuity monotone-functions
asked Jul 26 at 17:53
user63858
183
edited Jul 26 at 17:55
asked Jul 26 at 17:53
user63858
183
asked Jul 26 at 17:53
user63858
183
asked Jul 26 at 17:53
user63858
183
183
1
Something that jumps out to me immediately is that you haven't actually shown that $f$ is monotonic in the neighborhood. In particular, you need to show that for any $x,y in (c- delta, c + delta)$, we have $x < y Rightarrow f(x) < f(y)$. All you have done is shown that this works when comparing points in the interval to $c$.
– Joe
Jul 26 at 18:04
@Joe oh right i just realised that. So the only way to show that the function is monotonically increasing is to prove that it has a positive derivative in that interval right?
– user63858
Jul 26 at 18:11
No not necessarily. Often times it's easier to show a function is monotone directly from the definition as opposed to finding the derivative. See the Cantor Function for example.
– Joe
Jul 26 at 18:17
add a comment |Â
1
Something that jumps out to me immediately is that you haven't actually shown that $f$ is monotonic in the neighborhood. In particular, you need to show that for any $x,y in (c- delta, c + delta)$, we have $x < y Rightarrow f(x) < f(y)$. All you have done is shown that this works when comparing points in the interval to $c$.
– Joe
Jul 26 at 18:04
@Joe oh right i just realised that. So the only way to show that the function is monotonically increasing is to prove that it has a positive derivative in that interval right?
– user63858
Jul 26 at 18:11
No not necessarily. Often times it's easier to show a function is monotone directly from the definition as opposed to finding the derivative. See the Cantor Function for example.
– Joe
Jul 26 at 18:17
1
1
Something that jumps out to me immediately is that you haven't actually shown that $f$ is monotonic in the neighborhood. In particular, you need to show that for any $x,y in (c- delta, c + delta)$, we have $x < y Rightarrow f(x) < f(y)$. All you have done is shown that this works when comparing points in the interval to $c$.
– Joe
Jul 26 at 18:04
Something that jumps out to me immediately is that you haven't actually shown that $f$ is monotonic in the neighborhood. In particular, you need to show that for any $x,y in (c- delta, c + delta)$, we have $x < y Rightarrow f(x) < f(y)$. All you have done is shown that this works when comparing points in the interval to $c$.
– Joe
Jul 26 at 18:04
@Joe oh right i just realised that. So the only way to show that the function is monotonically increasing is to prove that it has a positive derivative in that interval right?
– user63858
Jul 26 at 18:11
@Joe oh right i just realised that. So the only way to show that the function is monotonically increasing is to prove that it has a positive derivative in that interval right?
– user63858
Jul 26 at 18:11
No not necessarily. Often times it's easier to show a function is monotone directly from the definition as opposed to finding the derivative. See the Cantor Function for example.
– Joe
Jul 26 at 18:17
No not necessarily. Often times it's easier to show a function is monotone directly from the definition as opposed to finding the derivative. See the Cantor Function for example.
– Joe
Jul 26 at 18:17
add a comment |Â
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2863647%2fsufficient-condition-to-show-that-fx-is-monotonic-in-some-neighbourhood-of-x%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
1
Something that jumps out to me immediately is that you haven't actually shown that $f$ is monotonic in the neighborhood. In particular, you need to show that for any $x,y in (c- delta, c + delta)$, we have $x < y Rightarrow f(x) < f(y)$. All you have done is shown that this works when comparing points in the interval to $c$.
– Joe
Jul 26 at 18:04
@Joe oh right i just realised that. So the only way to show that the function is monotonically increasing is to prove that it has a positive derivative in that interval right?
– user63858
Jul 26 at 18:11
No not necessarily. Often times it's easier to show a function is monotone directly from the definition as opposed to finding the derivative. See the Cantor Function for example.
– Joe
Jul 26 at 18:17