Other important contributions to mathematics that he made involved discontinuous functions and divergent series, differential equations, interpolation, the gamma function and elliptic functions.
12.
Indeed, there are many functions that are integrable but lack elementary antiderivatives, and discontinuous functions can be integrable but lack any antiderivatives at all.
13.
Though this result is not correct without additional conditions, Fourier's observation that some discontinuous functions are the sum of infinite series was a breakthrough.
14.
However, Lorenz curves representing discontinuous functions can be constructed as the limit of Lorenz curves of probability distributions, the line of perfect inequality being an example.
15.
In Kerner s theory, the probability of over-acceleration is a discontinuous function of the vehicle speed : At the same vehicle density, probability of over-acceleration in free flow is greater than in synchronized flow.
16.
This is done by using the so-called generalized gradient smoothing technique, with which one can approximate the gradient of displacement functions for certain class of discontinuous functions, as long as they are in a proper G space.
17.
Also, for indeterminate forms, couldn't we just look at the value of a discontinuous function at the discontinuity, such as tan ( 90?) ( that is + \ infty and / or-\ infty )?
18.
The assumption of continuity is important, as the discontinuous function f : x \ mapsto ( 1-x ) ^ {-1 }, for which every non-zero value has period 3, would otherwise be a counterexample.
19.
Then each function & fnof; " n " is continuous, but the sequence converges pointwise to the discontinuous function & fnof; that is zero on [ 0, 1 ) but has & fnof; ( 1 ) = 1.
20.
In this article, we will not be concerned with convergence issues; it is nice to note that all Lipschitz-continuous functions have a converging Fourier series expansion, and nice enough discontinuous functions have a Fourier series that converges to the function value at most points.
