Function differentiable on $(a,b)$ but not continuous on $ [a,b]$

Question

Is there any function $f$ which is differentiable on an open interval $(a,b)$ but is not continuous on (and also cannot be extended continuously to) the closed interval $[a,b]$?

@Anne You should tell us what attempts you have made to find a function like this — terrace, Nov 26 '14 at 14:35
"not continuous on $[a,b]$" and "can't be extended continuously to $[a,b]$" are not the same things - which one did you mean? (the answers here seem to suppose the first meaning)
an example of the latter kind: $sin(1/x)$ on $(0,1)$ — Dániel G., Nov 26 '14 at 15:26

score 35 · Answer 1 · edited Nov 26 '14 at 15:45

35

$$\Large\dot{}\!\!\underline{\qquad\qquad\qquad}\!\!\Large\dot{}$$

$$$$

edited Nov 26 '14 at 15:45

Did

279,727

answered Nov 26 '14 at 12:44

Zubin Mukerjee

17,832

What is it? Some sort of joke? – ajotatxe Nov 26 '14 at 12:45
4

It is supposed to be a graph of such a function :) – Zubin Mukerjee Nov 26 '14 at 12:46
._.$\phantom{2}$ – Upc Mar 30 '18 at 19:37

score 6 · Answer 2 · answered Nov 26 '14 at 13:50

6

$$f(x)=\frac1{(x-a)(x-b)},\qquad f(a)=f(b)=0.$$

answered Nov 26 '14 at 13:50

Martín-Blas Pérez Pinilla

41,546
4
46
89

score 4 · Answer 3 · answered Nov 26 '14 at 13:00

4

The easiest function I can think of is

$$f(x)=\begin{cases}\sqrt x&,\;\;x\in (0,1]\\{}\\18&,\;\;x=0\end{cases}$$

answered Nov 26 '14 at 13:00

Timbuc

34,191

@Timbuc How did you think of it (I mean, rationale)? – hola Nov 26 '14 at 14:53
@pushpen.paul, the square root is my box example of function defined and continuous on a certain interval but not differentiable in one point there. – Timbuc Nov 26 '14 at 15:10

score 2 · Answer 4 · answered Nov 26 '14 at 12:39

2

Consider $f(x)=x-\lfloor x\rfloor$ at any interval $(n,n+1)$ with integer endpoints.

answered Nov 26 '14 at 12:39

ajotatxe

65,084

score 0 · Answer 5 · answered Nov 29 '14 at 22:48

Differentiability implies continuity, but the intervals $(a,b)$ and $[a,b]$ were not the same; the first was open second was closed. This means at the points $a$ and $b$ it can be not continuous and it will still be differentiable on open $a,b$.

Thusly you can have a function that does what you said.

For example, you could have $f(a) = 5$ and $f(x) =2$ otherwise (when $x \neq a$). This function will have a discontinuity at $x=a$.

Function differentiable on $(a,b)$ but not continuous on $ [a,b]$

5 Answers5