4

I would like to see a function $f:[a,b]\to\mathbb{R}$ that is differentiable in $(a,b)$ but it is not continuous at least at one of the interval boundary points $a$ or $b$. Can you show me one?

This is a curiosity that would make me to see limits of Rolle theorem, because one of its hypothesis is that the function $f$ has to be continuous in the entire closed interval $[a,b]$, even if it could be differentiable only in the open $(a,b)$.

Thank you.

Martin Sleziak
  • 53,687
Nameless
  • 81
  • 2
    What is the question? – William M. Feb 23 '19 at 19:09
  • The first phrase. I edit to make the question clearer. – Nameless Feb 23 '19 at 19:10
  • 1
    I think the question is pretty clear: "why can we not relax the requirement of continuity on $[a,b]$ in Rolle's theorem?". – Patrick Stevens Feb 23 '19 at 19:11
  • @PatrickStevens I was truly unsure what the OP was intending to ask. The reply to my comment made even less sense. I am not sure how you have the ability to read peoples minds through what they write, but OK. – William M. Feb 23 '19 at 19:51
  • I can not understand how my simple answer "the first sentence is my question" could not make sense, but OK. – Nameless Feb 23 '19 at 19:54
  • 1
    Essentially if $f$ is not continuous at both $a$ and $b$ then knowing $f(a)=f(b)$ is not helpful information. But you need that to avoid strictly monotonic functions – Henry Feb 23 '19 at 23:50

2 Answers2

7

Consider $f(x) = x$ on $(0, 1]$, and $f(0) = 1$.

Patrick Stevens
  • 36,135
  • 1
    Who knows for what obscure reason I was imagining things incredibly more complex than that. Thank you very much. Sorry for the question that now seems very stupid to me. – Nameless Feb 23 '19 at 19:17
  • 3
    In first-year analysis, there are about five different counterexamples, all of them quite simple. Almost nothing you'll encounter will require really pathological counterexamples. – Patrick Stevens Feb 23 '19 at 19:19
2

Consider $f(x) = \frac{1}{x}$ in $[0, 1]$ and define $f(0) = 0.5$.

Then by the extreme value theorem - which is needed to make Rolle's Theorem work - since $f$ doesn't obtain a maximum, $f$ is not continuous on $[a,b]$.

Martin Sleziak
  • 53,687
Jossie Calderon
  • 391