Recent content by mike1988

actually I figured this out. Since || is a continuous, |lim(x-->0)(f(x))|= lim(x-->0)|f(x)| which is obvious from one of the theorems in my book. Thanks though!

I am trying to solve a question and I need to justify a line in which |lim(x-->0)(f(x))|≤lim(x-->0)|f(x)| where f is a continuous function. Any help?

Do you think I should go by contradiction instead? coz I am not really figuring out how to use this assumption here. Some elaboration would really help!

I guess I don't really get what this means (are you using the sing-change property of continuity?). I am really stuck! Thanks

At first I thought it was wrong, so was trying to find a counterexample. But I saw this question in a book asks to show that f(x)=0 for all x in R, which means it should be true. I was using Mean Value theorem and fundamental theorem of calc but I am not really heading anywhere with those...

So after using Mean Value theorem and FUndamental theorem of Calc, I got to a point where F'(c)=f(c) ≤ m - n. BUt this still doesn't seem to work enough. Is there anything crucial that I am missing? Thanks!

I am sorry. Here is the exact question: Let f : R to R be a continuous function, and suppose that definite integral from m to n |∫(m to n)f(x)dx|≤(n-m)^2 for every closed bounded interval [m, n] in R. Then is it the case that f(x) = 0 for all x in R?

Let f : R to R be a continuous function, and suppose that definite integral from m to n |∫(m to n)f(x)dx|≤(n-m)^2 for every closed bounded interval [m, n] in R. Then is it the case that f(x) = 0 for all x in R? I tried using fundamental theorem of calculus but got stuck, since I only got that...

Let f : R to R be a continuous function, and assume anti-derivative of f(x)dx from m to n≤ (n-m)^2 for every closed bounded interval [m,n] in R. Prove that f(x) = 0 for all x in R. I tried using fundamental theorem of calculus but got stuck. Any help/suggestion would be appreciated.