I am learning about differentiability of functions and came to know that a function at sharp point is not differentiable.
For eg.
$$f(x)=|x|$$
I could find out that $f(x)$ is not differentiable at $x=0$ because
$$\lim_{x\to 0^-}f'(x) \ne \lim_{x\to 0^+}f'(x) $$
This is all mathematical but I couldn't understand where the sharp point plays its role here ?
How sharp point makes these limits to evaluate different ?
Tell me more
×
Mathematics Stack Exchange is a question and answer site for
people studying math at any level and professionals in related fields. It's 100% free, no registration required.
|
|
|||||||||||
|
|
First remark: your $f$ is not differentiable (at $0$) because the limit $$ \lim_{h \to 0} \frac{|h|}{h} $$ does not exist. In general the limit of $f'$ is only a sufficient condition for differentiability. Be very careful, if you use it to disprove differentiability. Have you tried to sketch the graph of $f$? If so, you have seen that there is no tangent line to the graph at $0$, because of the sharp point. This is way to "understand" the rôle of the sharp point. But again, be careful: differentiability is a mathematical idea. The best way to understand it, is to understand it mathematically, according to the definition. Everything else may be misleading. |
|||||
|
|
A function is differentiable at a point, $x_0$, if it can be approximated very close to $x_0$ by $f(x)=a_0+a_1(x-x_0)$. That is, up close, the function looks like a straight line. A kink, like you see in $|x|$ at $x=0$, is where the graph of $|x|$ does not look like a straight line. Rather than look at $$ \lim\limits_{h\to0^+}f'(x+h)\quad\text{and}\quad\lim\limits_{h\to0^-}f'(x+h)\tag{1} $$ w should look at $$ \lim\limits_{h\to0^+}\frac{f(x+h)-f(x)}{h}\quad\text{and}\quad\lim\limits_{h\to0^-}\frac{f(x+h)-f(x)}{h}\tag{2} $$ If $f$ is continuous and the limits in $(1)$ exist and are equal, then $f'(x)$ is equal to those limits. However, if $$ f(x)=x^2\sin(1/x)\tag{3} $$ then the limits in $(1)$ do not exist for $x=0$, yet $f'(0)=0$. However, by definition, if and only if the limits in $(2)$ exist and are equal, does $f'(x)$ exist and equal to those limits. |
||||
|
|
|
${}{}{}{}{}{}{}{}{}{}{}{}{}{}{}$ |
|||||
|
