Does anyone know an example of a real function $f$ continuous in $[a,b]$ and differentiable in $(a,b)$ such that the lateral derivatives $$ \lim_{h \to a^{+}} \frac{f(x+h)- f(x)}{h} \quad \text{and} \quad \lim_{h \to b^{-}} \frac{f(x+h)- f(x)}{h}$$ don't exist?
$f$ continuous in $[a,b]$ and differentiable in $(a,b)$ without lateral derivatives at $a$ and $b$
3
$\begingroup$
real-analysis
3 Answers
5
$$f(x) = \sqrt{x-a} + \sqrt{b-x} \,\,\,\,\,\, \forall x \in [a,b]$$
-
0very nice! how did you come up with this? – 2012-12-25
-
2Based on your knowledge, the proper answer to the question is "Yes"$ – 2012-12-25
-
2@user32240 I guess that is a harder question to answer :). Essentially, I wanted a continuous function whose derivative didn't exist at a point. Say this function is $g(x)$ i.e. it is continuous for $x \geq a$ and differentiable for $x>a$. Once I have this, I can make use of this function to construct another function, which does not have derivatives at $a$ and $b$ by setting $f(x) = g(x) + g(b+a-x)$. – 2012-12-25
2
$f(x)=(x-a)\sin(1/(x-a))+(x-b)\sin(1/(x-b))$
1
A semicircle centered at the midpoint of $[a,b]$ with radius $(b-a)/2$ is an example: $$f(x)=\sqrt{ [(b-a)/2]^2 - (x-[(a+b)/2])^2 }.$$