Continuous and differentiable functions

[member 587159]

"If a function can be differentiated, it is a continuous function"

By contraposition: "If a function is not continuous, it cannot be differentiated"

Here comes the question: Is the following statement true?

"If a function is not right(left) continuous in a certain point a, then the function has no right(left) derivative in that point"

Thanks in advance

 

[RUber]

I have never heard of a right or left derivative, but I suppose the logic is right.
Think of the limit form definition of the derivative. If you look at:
$\lim_{x\to a^+}\frac{f(x)-f(a)}{x-a}$ as the definition of a right derivative at a, then clearly if there is a dicontinuity on the right side of a there can be no derivative.

 

[micromass]

Please define the term "left continuous" and "left derivative".

 

[member 587159]

Please define the term "left continuous" and "left derivative".

I translated this freely to English. I supposed it was clear when I used "left continuous and left derivative" ,

left continuous; lim x>a- f(x) = f(a)
left derivative; lim x>a- (f(x) - f(a))/(x-a)
Those are the definitions I know for functions R -> R

 

[mathwonk]

just use the same argument as for 2 sided limits.

 

[pwsnafu]

I have never heard of a right or left derivative, but I suppose the logic is right.

Please define the term "left continuous" and "left derivative".

Technically, left and right differentiability refer to semi-differentiability, which is weaker than normal differentiability.