Note the definition given. The discrete time function X is defined as a sum of values of two continuous functions evaluated at discrete points or sequences.  So the time function is the restriction of a continuous function defined on sequences. 

Consider f(x) = x^2 - continuous function but you can define a discrete time function as x(k+1) := f(x(k)) = (x(k))^2 where x(k) is some sequence. 

If discreteness satisfies \lim \Delta t \rightarrow 0 there will be little problems. But, I am afraid that nature refuses to work that way in support of any model. 

You are heavily misunderstanding the things. f and g are continuous functions of x. In discrete-time systems, the state usually varies continuously.

A clarification on the commnet by Pavel: In discrete-time systems, the state varies usually  since the state is defined for all time and has a time-derivative . The discretization consist of picking up samples of the continuous-time function to define difference equations at sampling times or other alternative descriptions.  In many situations the  (discrete-time) system is a particular case of a continuous-time system where the input is piece-wise constant ( generated by a zero -order-hold) or some more complicated inputs ( first- order hold or fractional or generalized  holds) .  However, the system can be instead  a digital ( or discrete ) system in the sense that it is only defined at sampling points with no relation to a primary continuous-time system. I think  that in both cases it is nonsense to speak about smoothness  when dealing with the discrete description. However, in the first case , there is typically a continuous- time  differentiable function defined the state , then defined for all time , and, for this description , we can speak about  its smoothness.

I think there is an invisible down voter who does not know the very question asked. 

I am sure this statement of mine will be down voted. 

Dear Nalin,

The other possibility you might have to consider is to define smoothness in such a case when the function used,  f  is smooth ( infinitely many times differetiable or up to a required order ). It is indicated that the function f is continuous but did not mention about its smoothness, since a continuous function may not be differentiable of  first order or higher and hence not smooth. 

I think there is a terrorizing agent here in RG. This is a place of ideas where we openly communicate and there is no need simply to down vote a comment while there is a page to wrote a comment on it. 

My last comment is purely a fact of mathematics and I am not sure what the down vote on it  unless the aim is to terrorize. 

Please note that, smoothness of a function (i.e. f(.) or g(.)) has nothing to do with the discrete-time evaluation of that function. In other words, when it is assumed that f(.) is smooth, it means that the continuous-time function f(.) is smooth in its domain, though you may evaluate f(.) in some sampling instances.

It depend on the sampling time, become small it will become smooth.

Indeed but the positions of sample values, where they lie ( smooth continuous paths or otherwise) also determine their behaviors, how erratic or non erratic the dynamical system is in the discrete time scale, which is controlled by the behavior of the function f. 

There is a problem when some people define some concept, or just use some name, for some specific need and then other people try to find some general meaning.

There is no way I (we) can guess why they needed the smoothness in the discrete-time system, yet as you mentioned that f and g are smooth and they are only functions of x, then they ARE smooth. Period!  The discrete-time has nothing to do with this.

BTW, mentioning discrete-time is only to confuse us, at least at first reading.

My own first instinctive reply was "there can be no such a thing." 

If we know that there exists a continuous function having some known values at some equidistant points in a compact interval, and one requires the recovery of that function in the whole interval, one can use Bernstein polynomials. The sequence of these polynomials converges uniformly to our function on the given interval. Thus, polynomial approximation holds. The method works in several dimensions too, where the form of Bernstein polynomials is also known. If the points are not equidistant, one can add some other points, such that the new finer division of the interval to be close to an "equidistant" one.

The authors of the paper are correctly speaking about the smoothness of function f: R^n-->R^n (and some other functions). This is a function of n variables, thus its differentiability is a well-defined notion. It has no connection with any time (continuous or discrete). Function x (the state) is of type x: Z-->R^n, i.e. it is defined for discrete-time values. The composit function f(x(.)) appearing on the rigth hand side of the system's model is mapping from Z to R^n. You are right that it is meaningless to speak about its smoothness. But the paper doesn't do that.

Try to look at the paper again keeping in mind what are the domains of definition of the functions. 

I personally believe the metric for smoothness has to be defined first. Following that, one then can talk about smoothness of something.

If the discrete set is also closed, under some hypothesis on the function, we can apply Whitney extension theorem (seeWikipedia). Those hypothesis are related to the Taylor formula.

No, discrete-time functions are not smooth in the conventional sense. But they are inifinitely smooth in the generalized functions sense. Recall that the Dirac delta can be derived infinitely often in the generalized functions sense. A discrete-time function (or simply discrete function) is a linear combination of weigthed Diracs.

It might feel somehow strange that a Dirac comb which coincides with the sequence [..,1,1,1,1,1,1,..] is a smooth function (in the distributional sense). But recall that a Dirac comb (because it is periodic) can be developed into a Fourier series which is a weighted superposition of smooth functions (complex exponentials). This phenomenon can only be explained within the theory of generalized functions. Outside this theory, Fourier series remain a paradox.

Now, let me just mention that every (generalized or ordinary) function which is not smooth (infinitely differentiable) in the ordinary functions sense can be made smooth in the ordinary functions sense by convolving it with a Gaussian (which itself can be arbitrarily close at the Dirac delta). The result of this convolution is a function that approximates your original discrete-time function arbitrarily well and is smooth (can be differentiated infinitely often) in the ordinary functions sense. 

The distance between, with dimension is often relevant. The logistic map is well-known and taught in several courses of bifurcations, and of the kind in the question. We found a result for the first bifurcation, that I did not know, when introducing dimensions of the variable. The ratio was determined, not bifurcated.

https://www.researchgate.net/publication/312053900_Concepts_of_nco_related_to_cosmos_states_Feynman-collapse_and_bifurcations_in_the_logistic_map

By making the discrete intervals sufficiently small, the definition of a continuous function is accomplished.

"Discreteness" is actually a mathematical idealization which does not exist in the real world..

Every discrete function which arises in the real world is also smooth to some very small extent. Otherwise we would not be able to see line spectra. In line spectra (which are Dirac combs in generalized functions theory), every line is expected to be infinitely thin. In this way, we would not be able to "see" them. However, this is not true in reality. Every line has a "width" which arises due to some "regularization" process which counteracts the "discretization" process.

Article On the Reversibility of Discretization

The width of line spectra is described, e.g. in:

Pascal Jordan, Anschauliche Quantentheorie: Eine Einführung in die moderne Auffassung der Quantenerscheinungen, Springer, Berlin 1936.

https://en.wikipedia.org/wiki/Spectral_line

Another example is the fact, that "No wave is truly monochromatic, although some waves, such as those produced by lasers, are exceedingly close to being so". See: Mark Jarrell's Lectures on Electrodynamics, Chapter 7

http://www.phys.lsu.edu/~jarrell/COURSES/ELECTRODYNAMICS/Chap7/chap7.pdf