# Linked Questions

2answers
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### 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? ...
4answers
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0answers
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### 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 ...
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 ...
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 ...
1answer
277 views

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 $\displaystyle\lim_{x\to ... 1answer 253 views ### Show that if$\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)$... 1answer 316 views ### Show that if$\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$is ... 1answer 79 views ### Show$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$is ... 2answers 74 views ### if$\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 $1answer 91 views ###$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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