True/False: f'(a) Exists if f(x) is Continuous, Limit of f'(x) is c at x->a

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In summary, the conversation discusses the existence of f'(a) when given the information that f(x) is continuous and the limit of f'(x) as x approaches a is c. The person asking the question clarifies if the statement is always true, and the response explains how to use the definition of the derivative and the mean value theorem to determine the existence of f'(a). The conversation also mentions a theorem about the limit of composite functions, which may be used in the process.
  • #1
tsuwal
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Homework Statement


True/False:f(x) is continous, limit of f'(x) as x->a is c, then f'(a) EXISTS equals c

Homework Equations


The Attempt at a Solution


I know that if f'(a) exists the statement is true, but is it true that based on that information f'(a) exists?

Homework Statement


Homework Equations


The Attempt at a Solution

 
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  • #2


Look at the definition of the derivative. [itex]f'(a) = c[/itex] means that
$$\lim_{x \rightarrow a}\frac{f(x) - f(a)}{x - a} = c$$
Try applying the mean value theorem to
$$\frac{f(x) - f(a)}{x - a}$$
and see if you can conclude anything.
 
  • #3


yes, my teacher explained t that way but the last part of the demonstracion when he uses some theorem about the limit of compound functions with csi(x) is really confusing..
 
  • #4


Suppose [itex]x > a[/itex]. Does the mean value theorem apply to [itex]f[/itex] on the interval [itex][a, x][/itex]? If so, what does it say?
 

Related to True/False: f'(a) Exists if f(x) is Continuous, Limit of f'(x) is c at x->a

1. What does it mean for a function to be continuous?

A function is continuous if it has no breaks or gaps in its graph, meaning that the value of the function at a given point is equal to the limit of the function at that point.

2. How is the derivative of a function related to its continuity?

The derivative of a function measures the rate of change of the function at a given point. If the derivative exists at a point, then the function is continuous at that point.

3. Can a function be continuous but not have a derivative?

Yes, a function can be continuous but not have a derivative. This occurs when the function has a sharp corner or a vertical tangent at a point, making the derivative undefined.

4. What does it mean for the limit of a function's derivative to be a constant at a given point?

If the limit of a function's derivative is a constant at a given point, it means that the function is changing at a constant rate at that point. This is known as "differentiability" and is a necessary condition for the existence of the derivative at that point.

5. How does the existence of the derivative at a point affect the behavior of the function at that point?

If the derivative exists at a point, it means that the function is smooth and has a defined slope at that point. This allows us to make predictions about the behavior of the function near that point and to calculate its rate of change at that point.

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