Finding points of non-differentiability

[Ashu2912]

Hey friends! I am having a slight confusion as to finding the points of non differentiability of sum, product and composite of functions.
Consider the functions f and g. If f is differentiable on an interval and so is g, then this interval comes under the domain of f+g, and f+g is also differentiable on this interval. Similarly, for product... Now if we want to find the points of non-differentiability of f+g, we can't straightaway write all the points not included in the above interval, since we know that the function f+g is differentiable on that interval, but there is no comment about the differentiability at other points. Then how can we use this rule to find the points of non-differentiability of f+g or fg or fog?
For example, consider the function: |x||x|. Now this is fg, f:|x| & g:|x|. f, g are differentiable on all real numbers except 0. However, from the rule fg is differentiable on all R-{0}, which is true BUT NOT ONLY ON R-{0}, also at 0. Clearly we are unable to exploit the rule for the required purpose here!

 

[mathman]

There is no way. Let f be any function (even nowhere differentiable) and g = 1-f, then f+g is everywhere differentiable. Similar for product, let f (never 0) and g = 1/f.

 

[Deveno]

the above example should convince you that the set of "non-differentiable functions" (even if they are only non-differentiable at a finite set of points) is poorly behaved with respect to addition and multiplication (pointwise) of functions.

this suggests that perhaps they aren't very good objects of study, if we want to consider sums and products of functions (the sums and products have different properties then the functions we started out with).

functions involving absolute value can be problemmatic to differentiate, which is unfortunate, since it means that the "distance" function isn't differentiable along the line x = y. "jagged" objects (such as manifolds with corners, or a typical (line-connected) plot of stock prices over time, for example) don't lend themselves well to analysis, and can often display "unpredictable" behavior.

what is one to do?

well, we study the "nice" functions first. continuous is good, differentiable is better. we've gotten a lot of mileage from this "oversimplification", because many physical relationships seem to act in "well-behaved" ways. linear approximations (even though often vastly over-simplified) often work well "in the short-term", and we have come to understand those very well.

in other words: walking first, running and jumping later on.