Can Non-Differentiable Functions Affect Solving for Roots of a Function?

[eljose]

Let,s suppose we wish to calculate the roots of a function f(x) $$f(x)=0$$, of course you will say.."that,s very easy doc...just try Newton Method, fixed-point method or other iterative method"..the main "problem" we have is if f(x) includes non-differentiable functions such us the floor function [x] or the modulus of x |x| then how could we manage to solve it?..for example get the x values that satisfy $$g(x)-[g(x)]=0$$

the problem is that [g(x)] is not differentiable for certain values of x...how could we solve that?...

 

[Hurkyl]

Well, if you have a piecewise differentiable function, it seems an obvious thing to do is to work with each piece individually.

 

[HallsofIvy]

Or, if you really don't want to use the derivative at any point, fixed point or other methods will- the secant method or midpoint method, for example, still work.

 

[eljose]

But still we have the same problem or if x=r is a root of [g(x)]-f(x)=0 but at the point x=r the function [g(x)] has a discontinutiy...and for the secant method..is still valid for piecewise continuous and differentiable functions?, the method of fixed point, unless we are near we may have serious convergence problems to obtain the x so f(x)=x, another question..what would happen if f(x) is nowhere differentiable?..thanks.

 

[Edwin]

Is it possible to construct a complex-valued function g(w), w a complex variable, that generalizes the floor-function to the complex plane similar to the way that the gamma function of a complex variable z is a generalization of the factorial function to the complex plane? If so, and one can find it, one might ask "at what points in the complex plane is the generalized floor function g(w) complex differentiable in the complex plane?". It may be such that one can find an analytic continuation that extends the complex-valued function g(w) to the complex plane. Should you manage to get a closed form expression for g(w) at all points g(w) is defined, you can try to apply the argument principle to determine whether a region containing a portion of the real axis contains a zero of the function in question. One can then hone in on the zero of the function by interval halving. There are a lot of ‘ifs’ in all of this :) Is any of this plausible? If not, why not?

Inquisitively,

Edwin

 

[HallsofIvy]

To answer your specific question: "what if x= r satisfies f(r)= 0 but is a point of discontinuity of f?" Not much you can do! Since f can by any function at all for x not equal to r, information at any other x can't tell you anything about what happens at r.