Monday, 15 August 2016

calculus - What is the difference between differentiability of a function and continuity of its derivative?




I am sort of confused regarding differentiable functions, continuous derivatives, and continuous functions. And I just want to make sure I'm thinking about this correctly.



(1) If you have a function that's continuous everywhere, then this doesn't necessarily mean its derivative exists everywhere, correct? e.g., f(x)=|x| has an undefined derivative at x=0



(2) So this above function, even though its continuous, does not have a continuous derivative?



(3) Now say you have a derivative that's continuous everywhere, then this doesn't necessarily mean the underlying function is continuous everywhere, correct? For example, consider
f(x)={1x     x<02x     x0
So its derivative is -1 everywhere, hence continuous, but the function itself is not continuous?



So what does a function with a continuous derivative say about the underlying function?


Answer



A function may or may not be continuous.




If it is continuous, it may or may not be differentiable. f(x)=|x| is a standard example of a function which is continuous, but not (everywhere) differentiable. However, any differentiable function is necessarily continuous.



If a function is differentiable, its derivative may or may not be continuous. This is a bit more subtle, and the standard example of a differentiable function with discontinuous derivative is a bit more complicated:
f(x)={x2sin(1/x) if x00 if x=0
It is differentiable everywhere, f(0)=0, but f(x) oscillates wildly between (a little less than) 1 and (a little more than) 1 as x comes closer and closer to 0, so it isn't continuous.


No comments:

Post a Comment

real analysis - How to find limhrightarrow0fracsin(ha)h

How to find limh0sin(ha)h without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...