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DeadWolfe
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I'm only in high school, and I was wondering: Why are some functions not differentiable at certain points?
Originally posted by mathman
"Why" is a funny question. They just are. For example x.sin(1/x) does not have a derivative at x=0, even though it is continuous. There are more ccomplicated examples, like continuous functions with derivatives nowhere.
So how can [tex]f(x) = x\sin (x^{-1})[/tex] be a continuous function? I must lift my pencil because it is not defined at [tex]x = 0[/tex], is it?Originally posted by mathman
"Why" is a funny question. They just are. For example x.sin(1/x) does not have a derivative at x=0, even though it is continuous. There are more ccomplicated examples, like continuous functions with derivatives nowhere.
Doesn't there have to be an asymptote wherever a function is not defined? I may well be wrong here because I never really studied advanced math...Originally posted by matt grime
the reason xsin(1/x) does not come with a value at zero is cos of the 1/x inside the sin.
Originally posted by Chen
Doesn't there have to be an asymptote wherever a function is not defined? I may well be wrong here because I never really studied advanced math...
Originally posted by cookiemonster
Why should we have to impose the value of 0 at x = 0 on xsin(1/x) in order to make it continuous? The limit still exists, and that's what determines the continuity, right?
cookiemonster
Originally posted by cookiemonster
Yeah, I kinda realized that when I read my own definition. Thanks for the effort, though.
You have to admit, it's pretty impressive that somebody can throw out the definition of something and then change it in their own mind for no apparent reason not five minutes later.
cookiemonster
Originally posted by matt grime
There is an alternative definition (the 'proper one') that states: a function between topological spaces is continuous iff the inverse image of the open set is open.
As a map between R and R with the usual topology that map given is not continuous. If we were considering it as a map from R with the discrete topology to the two point set with the discrete topology (or R with the discrete topology) then it is continuous. This was just an observation so that no one can pull out the 'ha you've not considered this...' argument, and is just a cya, cos mathematicians are like that., just as we like to consider every possibility, just in case.
Originally posted by exekil
increadibly enough I understood some of that, it sounds very interesting, what kind of math does topology fall into? Did I just completely forgat about all of topology or do you lear that after calc.?
A function may not be differentiable at a point if it has a sharp corner or a discontinuity at that point. This means that the function does not have a well-defined slope at that point, making it impossible to calculate the derivative.
A function is differentiable at a point if it has a well-defined slope at that point, meaning that the derivative of the function exists at that point. This allows us to calculate the rate of change of the function at that point.
Yes, a function can be differentiable at some points but not others. This is because the function may have a well-defined slope at some points, but not at others where there are sharp corners or discontinuities.
To determine if a function is differentiable at a point, you can check if the function is continuous at that point and if the left and right-hand derivatives are equal. If these conditions are met, then the function is differentiable at that point.
If a function is not differentiable at a point, it means that the derivative of the function does not exist at that point. This can make it difficult to analyze the behavior of the function and can also affect its rate of change and curvature at that point.