Numerical methods root finding for function with one variable

Question

Fixed point iteration , bisection, newton , regula falsi, secant all methods are used for only continuous functions with one variable, aren't theY?

What can we think for numerically solving of one variable functions with noncontinous ?

Answer 1

Discontinuous function $f :\mathbb R \rightarrow \mathbb R$ occur in real life applications, but they tend to have only a finite number of discontinuities and these can be examined one by one. Standard methods can be used away from these points of discontinuity. An example could be the initial condition for a heat equation. Here the temperature can have a discontinuity if you put a piece of ice up against a piece of heated metal.

Functions $f : \mathbb R \to \mathbb R$ with more than a finite number of discontinuities occur in real analysis. Here is less meaningful to consider the concept of a zero, because two functions are equivalent if they agree everywhere except on a set of measure zero. In particular, they are equivalent if they agree everywhere except on a finite number of points. You can redefine your function, so that it has an extra zero anywhere you like and it will still represent the same equivalence class.