Differentiable functions proof

In summary, if a function g is differentiable at some point c in the interval (a,b) and its derivative g'(c) is not equal to 0, there exists a delta greater than 0 such that g(x) is not equal to g(c) for all x in the neighborhood of c, excluding c, and intersecting with the interval (a,b). This can be shown using the definition of a limit and choosing an appropriate value for epsilon.
  • #1
dancergirlie
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0

Homework Statement


Consider a function g : (a, b)-->R. Assume that g is differentiable at some point c in
(a,b) and that g'(c) is not = 0. Show that there is a delta > 0 so that g(x) is unequal to g(c) for all x in V_delta(c)\{c}intersect(a,b)


Homework Equations





The Attempt at a Solution



Alright, this is what I tried so far:

Since g is differentiable at c that means that

g'(c)=lim_x-->c (g(x)-g(c)/x-c)

since we are assuming that g'(c) is unequal to 0 that means that
lim_x-->c (g(x)-g(c)/x-c) is unequal to zero and thus g(x)-g(c) is unequal to zero
and therefore g(x) is unequal to g(c).

I don't really know where the delta comes in though, am I supposed to use the definition of a limit to show that there exists an epsilon greater than zero so that for |g'(c)-L|<epsilon?

Any help would be great!
 
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  • #2
dancergirlie said:

Homework Statement


Consider a function g : (a, b)-->R. Assume that g is differentiable at some point c in
(a,b) and that g'(c) is not = 0. Show that there is a delta > 0 so that g(x) is unequal to g(c) for all x in V_delta(c)\{c}intersect(a,b)


Homework Equations





The Attempt at a Solution



Alright, this is what I tried so far:

Since g is differentiable at c that means that

g'(c)=lim_x-->c (g(x)-g(c)/x-c)

since we are assuming that g'(c) is unequal to 0 that means that
lim_x-->c (g(x)-g(c)/x-c) is unequal to zero and thus g(x)-g(c) is unequal to zero
and therefore g(x) is unequal to g(c).

I don't really know where the delta comes in though, am I supposed to use the definition of a limit to show that there exists an epsilon greater than zero so that for |g'(c)-L|<epsilon?

Any help would be great!

The fact that [itex]\lim_{x\to c} (g(x)-g(c))/(x-c)= g'(c)[/itex] is not 0 does NOT imply that (g(x)- g(c))/9x-c) is not 0! It does mean that you can take [itex]\epsilon[/itex] to be |g'(c)/2|[/itex] and then, for some [itex]\delta> 0[/itex] if [itex]|x-c|< \delta[/itex], [itex]||g(x)- g(c))/(x-c)- g'(c)|< |g'(c)|/2[/itex]. Write that as [itex]-|g'(c)|/2< (g(x)- g(c))/(x-c)- g'(c)< |g'(c)|/2[/itex] and see where that leads.
 

FAQ: Differentiable functions proof

What is a differentiable function?

A differentiable function is a mathematical function that can be differentiated, or have its slope or rate of change calculated, at every point in its domain. This means that the function is smooth and continuous, with no sharp corners or breaks.

How do you prove that a function is differentiable?

To prove that a function is differentiable, you must show that it has a defined derivative at every point in its domain. This can be done by using the definition of differentiability, which involves finding the limit of the function's slope as the input approaches a specific point.

What is the difference between a differentiable and a non-differentiable function?

The main difference between a differentiable and non-differentiable function is that a differentiable function has a well-defined derivative at every point in its domain, while a non-differentiable function does not. This means that a differentiable function is smooth and continuous, while a non-differentiable function may have sharp corners or breaks in its graph.

Can all functions be differentiable?

No, not all functions can be differentiable. A function must have a well-defined derivative at every point in its domain to be considered differentiable. This means that functions with sharp corners, breaks, or discontinuities in their graph are not differentiable.

What is the importance of differentiable functions in mathematics and science?

Differentiable functions are important in mathematics and science because they allow us to model and analyze continuous and smooth phenomena. They are used in fields such as calculus, physics, economics, and engineering to understand the behavior and relationships between variables in a system.

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