consider: f: R R defined as:
f(x) = if
f(0) = 0.
prove f is not differentiable at 0.
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section A
going by the definition of differentiability, i understand it has to be shown that
= .
DOES NOT EXIST
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- i attempted computing the left and right hand limits but i dont think that is the right method?
section B
i also know if a function is differentiable at 0 then it has to be continuous at 0, which sin(1/x) is not. but i don't have a good idea on how to prove that sin(1/x) is not continuous at 0
??
also, would it be sufficient to just show that sin(1/x) is discontinuous at 0 to prove it is not differentiable at 0?
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please provide assistance greatly appreciated. once again, i'm assuming this must be simple and straight forward for an advanced mathematician
thnx captain and perfecthacker.
ok this is what i understood from what you said perfecthacker. here's my proof from both your help..
you used the sequential definition of continuity? which implies that:
continuity at 0 = limit at 0 exists and equals f(0)?
PROOF
sin(1/x) is continuous at if for any sequence R, such that , we have
now let and
so
if we consider, we have:
= = = 0
but, if we consider, we have:
= = = 1
we see there are points arbitrarily close to 0 where sin(1/x) = 1.
so this means the limit at 0 doesn't exist
as the limit doesnt exist. the function is not continuous. this implies that it is not differentiable at 0. Q.E.D
is this completely correct? and is there anything else i need to add onto this proof? thanks