What are the implications of having different lateral derivatives at a point?

[Taturana]

I'm reading about lateral derivatives...

I know that a function is said derivable on a point if the lateral derivative coming from left and the lateral derivative coming from right are both equal at that point. f'(a) exists only if both f'+(a) and f'-(a) exists and are equal... right?

Ok, now the book shows an example of a function that is not derivable at a point (so it has a "break" in its graph)... so the lateral derivative coming from the left at that point is different from the lateral derivative coming from the right... the book also says that for this function, at this point, there are two possible tangent lines...

My question is: if the lateral derivatives at a point of a function are different, does that implies that there will be two possible tangent lines? ALWAYS? Does this have a proof? Can someone show me the proof?

Thank you

 

[Mark44]

I'm reading about lateral derivatives...

The usual terminology that I'm more familiar with is "one-sided" derivative.

I know that a function is said derivable on a point if the lateral derivative coming from left and the lateral derivative coming from right are both equal at that point. f'(a) exists only if both f'+(a) and f'-(a) exists and are equal... right?

Yes. Since f'(a) is defined in terms of a (two-sided) limit, f'(a) exists if and only if both one-sided limits exist and are equal.

Ok, now the book shows an example of a function that is not derivable at a point (so it has a "break" in its graph)...

A function can be continuous at a point but not differentiable (we don't say "derivable") there. A very simple example of a function that is continuous everywhere but not differentiable at x = 0 is f(x) = |x|. If x > 0, f'(x) = 1. If x < 0, f'(x) = -1.

so the lateral derivative coming from the left at that point is different from the lateral derivative coming from the right... the book also says that for this function, at this point, there are two possible tangent lines...

My question is: if the lateral derivatives at a point of a function are different, does that implies that there will be two possible tangent lines? ALWAYS? Does this have a proof? Can someone show me the proof?

I don't see how it could be otherwise. The derivative (or one-sided derivative) gives you the slope of the tangent line at the point. The slope of the tangent line can be thought of as the limit of the slopes of secant lines between (a, f(a)) and (a + h, f(a + h)) as h approaches 0. If h is restricted to positive numbers, you have a right-side limit; if h is restricted to negative numbers, you have a left-side limit.

 

[Taturana]

The usual terminology that I'm more familiar with is "one-sided" derivative. Yes. Since f'(a) is defined in terms of a (two-sided) limit, f'(a) exists if and only if both one-sided limits exist and are equal.
A function can be continuous at a point but not differentiable (we don't say "derivable") there. A very simple example of a function that is continuous everywhere but not differentiable at x = 0 is f(x) = |x|. If x > 0, f'(x) = 1. If x < 0, f'(x) = -1.



I don't see how it could be otherwise. The derivative (or one-sided derivative) gives you the slope of the tangent line at the point. The slope of the tangent line can be thought of as the limit of the slopes of secant lines between (a, f(a)) and (a + h, f(a + h)) as h approaches 0. If h is restricted to positive numbers, you have a right-side limit; if h is restricted to negative numbers, you have a left-side limit.

Sorry for the wrong terminology I used, the book I'm reading is in portuguese, haha ;P

Okay, so at a non-differentiable point we have two possible tangent lines because we have also two secant lines (one that crosses the graph of left and one that crosses the graph of right)...

Then can I say that at a non-differentiable point of a function we have two slopes? Or there is no slope at a non-differentiable point (and I can only say that I have a slope for a right-sided limit and another for a left-sided limit)?

When a function is non-differentiable at a point, what does that mean? Does that mean that I can only specify at what rate y is increasing (compared to x) when I specify if I'm coming from left (left-sided limit) or right (right-sided limit)?

 

[Mark44]

Sorry for the wrong terminology I used, the book I'm reading is in portuguese, haha ;P

Okay, so at a non-differentiable point we have two possible tangent lines because we have also two secant lines (one that crosses the graph of left and one that crosses the graph of right)...

Then can I say that at a non-differentiable point of a function we have two slopes? Or there is no slope at a non-differentiable point (and I can only say that I have a slope for a right-sided limit and another for a left-sided limit)?

If a function is not differentiable at some point, then the derivative doesn't exist there. The slope of the tangent line is not defined there. Functions that are continuous but not differentiable at some point, have cusps (as does f(x) = |x| at x = 0).

Using that function as an example, the value of the left-side derivative at 0 is -1. The value of the right-side derivative at 0 is +1. Since they are different, the derivative at 0 doesn't exist.

When a function is non-differentiable at a point, what does that mean? Does that mean that I can only specify at what rate y is increasing (compared to x) when I specify if I'm coming from left (left-sided limit) or right (right-sided limit)?

It means that the derivative doesn't exist at that point. If the function is not differentiable at some point, but the left- or right-derivatives exist, then yes, you would have to specify those when talking about the rate of change of y with respect to x.

 

[Taturana]

If a function is not differentiable at some point, then the derivative doesn't exist there. The slope of the tangent line is not defined there. Functions that are continuous but not differentiable at some point, have cusps (as does f(x) = |x| at x = 0).

Using that function as an example, the value of the left-side derivative at 0 is -1. The value of the right-side derivative at 0 is +1. Since they are different, the derivative at 0 doesn't exist.
It means that the derivative doesn't exist at that point. If the function is not differentiable at some point, but the left- or right-derivatives exist, then yes, you would have to specify those when talking about the rate of change of y with respect to x.

Thank you again, now I understand =D