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Is it possible to have a function whose instantaneous rate at every X are different to each other such that there are no pattern of gradual change between them but the definition of derivative fails to acquired the rate at every X? such that the derivative at

X1=1 =7

X2=1.0000001 =-987

X3=1.0000002 =0.0089

The definition of derivative would fail since it expects that the neighbor instantaneous rate at X2 assumes the it is infinitely close to it which means it gradually change to X2 but it's not the case.A function who has an instantaneous rate at every point but its instantaneous rate to every point are connected such that no gradual change appears not even a single interval.Would a function exist? and why not?

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  • $\begingroup$ ... come again? $\endgroup$ – Clarinetist Oct 24 '14 at 4:28
  • $\begingroup$ It looks to me as if the question is whether, if $f$ is a differentiable function, the derivative $f\,'$ also has to be continuous. $\endgroup$ – Lubin Oct 24 '14 at 4:32
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    $\begingroup$ @Lubin: it depends on whether you read the question title or the question body. In any case whether the question is about a continuous function whose change is so irregular that it nowhere has a derivative, or about a function that does have a derivative in each point but which is itself not continuous, such things are possible (and in fact common!). However you need to sharpen your idea of what a function and being differentiable are to appreciate these examples; an explanation at the vague "rate of change" level of precision seems very hard to give. $\endgroup$ – Marc van Leeuwen Oct 24 '14 at 4:45
  • $\begingroup$ You seem to think that $10^{-7}$ is automatically small. It is not. Functions and their derivatives can do anything in that interval. It is not hard to find a nice smooth function that meets your specs (aside from $1=7$ and the like). We just need to make sure the $h$ in the definition of derivative is smaller than $10^{-7}$, but that is OK. $\endgroup$ – Ross Millikan Oct 24 '14 at 4:45
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There are continuous, but nowhere differentiable functions. That answers your second question.

The derivative of a function can exist, but be discontinuous at some points. The standard example is $x^2\sin(1/x)$ when $x\ne 0$, and $f(0)=0$. This is differentiable everywhere, but the derivative is not continuous at $x=0$.

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