Hello all, I have the following question: what conditions must a discontinuous $f$ satisfy for the $\int f(x) \ dx$ to be continuous?

I would say the set of discontinuities must have measure zero.

Am I missing something else?

if $f$ is discontinuous then can we perform the integration $\int f dx ?$

yes perhaps i have seen the characteristic functions integrated this way in measure theory

@BAYMAX I didn't answer that question but posted a comment:

@SimplyBeautifulArt: And then I noticed that you had posted this wrong answer many months ago. So many reasons why it is wrong, and it's best that you delete it.

@user21820 Yeah sure :P

@SimplyBeautifulArt That was quick. Didn't know you were around. Thanks!

@user21820 =P You aren't in the rooms I usually find you in...

Your answer is related to implicit function theorem right @user21820

and also looks interesting

@BAYMAX It's a vast generalization.

I will read it

means?

Well it's a generalization in the sense that derivatives are treated in terms of a parameter, rather than forcing one of the coordinates to be a parameter. So sometimes curves that don't have any ordinary derivative at a point have a derivative under my framework, such as the second example.

nice, i will definitely read it!

This makes intuitive yet rigorous proofs of the various rules in the style of Leibniz actually possible. Feel free to ask anything about it and I will clarify.

yup! sure

> I just love coming up with all these strange curves! =) – user21820 Feb 16 at 10:17

then @user21820 Fractals and Chaos theory gives ua lot of beautiful curves!

@BAYMAX Indeed, except they're not so much curves as patterns. =)

ok

You know about curves like continuous but nowhere differentiable curves, right?

it would be nice to study about the curves ! like theoir properties which may reveal a great deal of info!

$x \sin(1/x)$

?

If you have Graph I can just send you files for these curves.

sure but i wonder why you donot use Desmos?

Well, Graph allows recursive functions, which I use to construct the fractals. Can Desmos? I also use Geogebra, but like the crisp look of Graph.

Oh wow!

I realized I can't upload the file, but here are the functions:

I am dnldng graph

Press Ctrl+F to define functions. Then add the following.

g(x) = if(x<0 or x>1,g(x-floor(x)),abs(x-1/2)<err,1/2,x<1/2,x+g(2x)/2,1-x+g(2x-1)/2)

err = 2^-64

Then plot y=f(x) and y=g(x) to see these two well-known pathological graphs.

@user21820 Desmos can't do automatic recursion

I see thanks. Good to know. Anyway one has to be careful with Graph's numerical integration; it's not accurate in some pathological cases.

:P Not many good numerical integral calculators out there

And one more standard pathological curve:

y = x^2*sin(1/x^3)

Or more generally y = x^k*sin(1/x^n)?

f is an unbounded monotonic continuous function with zero derivative almost everywhere.

g is a bounded continuous function that is nowhere differentiable.

The third is a differentiable function whose derivative is not Riemann integrable.

@SimplyBeautifulArt For some k,n you get the above pathological behaviour. Too lazy to figure out which. =P

It's just those that are differentiable but whose derivative is unbounded near zero.

mhm