Linked Questions

7
votes
2answers
5k views

If a derivative of a continuous function has a limit, must it agree with that limit? [duplicate]

Suppose we have a continuous function $f : \mathbb{R} \to \mathbb{R}$. Suppose also that for a certain point $c$, $\lim_{x \to c} f'(x)$ exists. Must $f'(c)$ exist as well, and be equal to this limit? ...
6
votes
4answers
961 views

How to show $f'(0)$ exist and is equal to $1$? [duplicate]

Assume that $f$ be continuous on $\mathbb{R}$, $f'(x)$ exists for all $x\neq 0$, and $\lim_{x\rightarrow 0} f'(x)=1$. We need to show $f'(0)$ exist and is equal to $1$. $f'(0)=\lim_{x\rightarrow 0}\...
5
votes
4answers
237 views

Suppose $f$ is differentiable on $\mathbb{R}$ and that $\lim_{x \rightarrow 0} f'(x)=L$. May we conclude that $f'(0)=L$ [duplicate]

This seems like another good question for consideration. I think the answer is yes just because I cannot think of a way to make it break down because the domain is defined for all of $\mathbb{R}$. Any ...
3
votes
2answers
3k views

Showing that if $\lim\limits_{x \to a} f'(x)=A$, then $f'(a)$ exists and equals $A$ [duplicate]

Let $f : [a; b] \to \mathbb{R}$ be continuous on $[a, b]$ and differentiable in $(a, b)$. Show that if $\lim\limits_{x \to a} f'(x)=A$, then $f'(a)$ exists and equals $A$. I am completely stuck on it....
3
votes
1answer
328 views

$\lim_{x\to c}f'(x)=L$ implies $f'(c)=L$ [duplicate]

Let $f:[a,b]\to\mathbb{R}$ be differentiable on $[a,b]$ and let $c \in(a,b)$. Suppose that $\lim_{x\to c}f'(x)=L$ some $L \in\mathbb{R}$. Without using L'Hospital's Rule, prove that $f'(c)=L$. Hint: ...
4
votes
1answer
181 views

$\lim_{x \rightarrow 0} f'(x) = L$ exists, does it follow that $f'(0)$ exists? [duplicate]

The question is the following: Assume $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous, and for all $x \neq 0$, $f'(x)$ exists. If $\lim_{x \rightarrow 0} f'(x) = L$ exists, does it follow that $f'...
2
votes
0answers
1k views

Derivative exists when limit of derivative exists [duplicate]

Let $f$ be continuous on $(a,b)$ and let $c\in(a,b)$. Suppose $f'(x)$ exists for all $x\in(a,b)\backslash\{c\}$ and $\lim_{x\rightarrow c}f'(x)$ exists. Prove that $f'(c)$ exists. I don't know what ...
2
votes
3answers
112 views

Continuity of the derivative at a point given certain hypotheses [duplicate]

Suppose that $h$ is continuous on $[a,b]$ and differentiable on $(a,b)$, and that $c \in (a, b)$. Suppose also that $\lim\limits_{x \to c} h'(x)$ exists. Prove that $h'$ is continuous at $c$. I ...
1
vote
2answers
106 views

If $\lim_{x\to0}f'(x)=l$, then is it true that $f'(0)=l$. [duplicate]

Let $f$ be a differentiable function on an interval containing zero. If $$\lim_{x\to0}f'(x)=l$$ then is it true that $f'(0)=l$. If $f'$ is a continuous function, then of course it is true. But what ...
3
votes
1answer
277 views

Analysis proof mean value theorem [duplicate]

Let $f$ be a function which is continuous on an open interval $I$. Let $c$ be a point of $I$. Suppose that $f$ is differentiable at every point of $I$ other than $c$, and that $\displaystyle\lim_{x\to ...
1
vote
1answer
253 views

Show that if $\lim_{x\rightarrow a}f'(x)=A$, then $f'(a)$ exists and equals $A$. [duplicate]

Let $f:[a,b]\rightarrow\mathbb{R}$ be continuous on $[a,b]$ and differentiable on $(a,b)$. Show that if $\lim_{x\rightarrow a}f'(x)=A$, then $f'(a)$ exists and equals $A$. Use the definition of $f'(a)$...
2
votes
1answer
316 views

Show that if $\lim_{x\to a}f'(x) = A$ then $f'(a)$ exists and equals A [duplicate]

I ran across this problem in my Analysis class and can't seem to come up with a good solution. Here's the question: Show that if $\lim_{x\to a}f'(x) = A$ then $f'(a)$ exists and equals $A$. $f$ is ...
1
vote
1answer
79 views

Show $f$ is differentiable at the endpoint with $a$ with $f'(a)=c$ [duplicate]

Let $f: [a,b] \rightarrow \mathbb{R}$ continous on [a,b] and differentiable on $(a,b)$. Asume there exist some $c \in \mathbb{R}$ so $f'(x) \rightarrow c$ for $x \rightarrow a+$. Show $f$ is ...
-2
votes
2answers
74 views

if $\lim \limits_{x \to x_0} f'(x) = L$ then $f'(x_0) = L $ [duplicate]

Let $ f $ be a function that is differentiable on a deleted neighborhood of $x_0 ∈ R$ and continuous at $x_0$. Show that if $\lim \limits_{x \to x_0} f'(x) = L$ then $f'(x_0) = L $
3
votes
1answer
91 views

$f$ is differentiable. If $\lim_{x \to c}f'(x)$ exists, then this limit must be $f'(c)$. [duplicate]

Please prove: Let $f:(a,b) \to R$ be differentiable function, and let $c \in (a,b). $ If $\lim_{x \to c}f'(x)$ exists and is finite, then this limit must be $f'(c)$. I tried doing it directly but ...

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