Function equal up to nth order, diffbility

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Homework Statement

Two functions f,g:R->R are equal up to n^th order at if

lim h-->0 [f(a+h) - g(a+h) / hⁿ ]= 0

Show that f is differentiable at a iff there is a function g(x)=b+m(x-a) such that f and g are equal up to first order at a.

Homework Equations

f is differentiable at a if the following limit exists:

lim h-->0 [f(a+h) - f(a) / h]

The Attempt at a Solution

Direction 1:
=>

Assume f is differentiable. Show there exists a function g(x)=b+m(x-a) such that f and g are equal up to first order at a.

So, lim h-->0 [f(a+h) - f(a) / h] exists and equals some L.

=>lim h-->0 [f(a+h) - f(a) - g(a+h) + g(a+h)/ h]
=>lim h-->0 [(f(a+h)- g(a+h)) - f(a) + g(a+h)/ h]

this implies that we want to show

lim h-->0 [g(a+h) - f(a)/h = L]

I'm not sure where to go from here.


Direction 2:
<=

Assume that lim h-->0 [f(a+h) -g(a+h) / h = 0]
show lim h-->0 [f(a+h) - f(a) / h] is equal to some constant.


Take lim h-->0 [f(a+h) -g(a+h) / h = 0]:

=> lim h-->0 [f(a+h) -g(a+h) -f(a) + f(a) - g(a) + g(a) / h = 0]

=> lim h-->0 [(f(a+h)-f(a)) -(g(a+h) - g(a)) + (f(a) - g(a))/ h = 0]

but lim h-->0 [f(a+h)-f(a)/h] is just f'
and lim h-->0 [g(a+h) - g(a)/h] is -m
so, we just need to show that
f(a) = g(a) and then lim h-->0 [f(a) - g(a)/h] = 0 and f' = m
Not sure how to do that.

That's all.

Thanks for your help.

 

[kekido]

There exists g(x)=m+b(x-a) such that f and g are equal to the first order, so you need to find such g (i.e., the value of m and b) according to the diff'bility of f.

 

[redyelloworange]

for both parts?

 

[HallsofIvy]

No, kekido said "such that f and g are equal to the first order" so he is talking about "if there exist g= m(x-a)+b such that f and g are equal to first order, then f is differentiable at x=a."

However, the theorem as stated is not true. For example, if f(x)= 2x if x is not 0, 1 if x= 0, and g(x)= 2x, we have f(0+h)= 2h for all non-zero h, g(0+h)= 2h for all h so
[tex]lim_{x\rightarrow a}\frac{f(0+h)-g(0+h)}{h}= 0[/itex]
but f is not differentiable at n.

You have to have some statement about what happens at x= a.