idle math musing

[learnedax]

Suppose there is a function f(x), which can be differentiated an infinite number of times. Suppose that a certain f(x₀) evaluates to 0, and so do f'(x₀), f''(x₀), etc. for every finite number of differentials. Now suppose that we can determine what the Nth differential of f(x) is in the limit as N goes to infinity, and that this limit differential is non-zero at x₀.

Essentially this would mean all finite rates of change would be nil, but the function would still not actually be flatlined (though unless the limit differential is infinite it'd take infinitely long for f(x) to show any change).

Is there any reason such a function could not exist? Do you know of any functions that have similar characteristics?

Comments

[londo.livejournal.com]

I'm unaware of anything which could remain exactly zero through every finite number of iterations, but becomes non-zero in the infinite case.

[mesatchornug.livejournal.com]

yeah. f'(0) is, by definition, 0.

and df(0) is likewise a constant.

[learnedax.livejournal.com]

Er, as you have stated that it is not really true. If f(x) is sin(x), then f(0) = 0, f'(0) = 1.