Since we know that in a good linear approximation, L ( x ) = f...

ziphumulegn

ziphumulegn

Answered

2022-07-09

Since we know that in a good linear approximation, L ( x ) = f ( a ) + f ( a ) ( x a ) .. But what if f ( a ) does not exist? How to prove that if a function has a good linear approximation, then it must be differentiable?

Answer & Explanation

Dobermann82

Dobermann82

Expert

2022-07-10Added 15 answers

One definition of having a good linear approximation for f at a is
f ( x ) = f ( a ) + c ( x a ) + ϵ ( x a )
where ϵ ( x a ), the error term, is a function that satisfies the following
lim x a ϵ ( x a ) x a = 0
Then, that f is differentiable basically follows immediately, by subtracting f ( a ), dividing by x−a and taking the limit as x a showing that the function is differentiable at a, with value c.
f ( x ) = f ( a ) + c ( x a ) + ϵ ( x a ) f ( x ) f ( a ) = c ( x a ) + ϵ ( x a ) f ( x ) f ( a ) x a = c + ϵ ( x a ) x a
and thus taking the limit
lim x a f ( x ) f ( a ) x a = c + lim x a ϵ ( x a ) x a f ( a ) = c

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