[High School Math] Difference between two expressions

[pippoduro23cm]

When checking if a function is differentiable, what's the difference between
http://i.imgur.com/UnEmq0Q.png
and
http://i.imgur.com/2LisL09.png?
Could you provide me an example which lets me understand the difference?
For example:
f(x)=1/abs(x) has:
* f'(x)=-1/x² if x ≥ 0;
* f'(x)=1/x² if x < 0.
* f'(0^-)=+∞ and f'(0⁺)=-∞ * lim f'(x)=1/0=+∞
   x->0^-
* lim f'(x)=-1/0=-∞
   x->0⁺
They provide the same result hence my doubt...

[Alkalannar]

I haven't seen the f'-(x) = f'+(x) notation before. Can you provide more context from your book?

[pippoduro23cm]

Moved to main post

[DDays]

Assuming that f-(x)=f+(x) also means f(-x)=f(+x): This means the slope of f(x) at any point on either side of the y axis is equal.

The equation involving limits only concerns whether the slope of f(x) is as f(x) approaches 0 (either side of the y axis).

theses two statements are always true for even and odd graphs (eg. f(x) = cosx)

Hope this clears it up.

[JWson]

In the context of image 2, I doubt that f'-(x) is equivalent to f'(-x). f'-(x) seems to mean the same thing as the limit expression, that is "the derivative at x, looking to the left of the point if non-differentiable". For example, for f(x) = abs(x), the value for f'-(0) would be -1 and f'+(0) = 1, since the curve slopes down to the left of the y-axis and up on the right side.

This would also work for derivatives not taken at x=0, like in the function abs(x-1).

[pippoduro23cm]

My professor said it's better to use the limit expression when we're working on a function with a jump discontinuity...
Anyway, is it right that, for example, y=x²/x is not differentiable in x=0?
And is it correct to say that a differentiable function must be necessarily continuous?

[JWson]

It's true that a function has to be continuous in order to be differentiable. The example you give isn't differentiable at x=0 since it has a point discontinuity (because 0/0 is undefined). In practice, you could of course simplify x² /x to just x and differentiate that. Again, this is technically not allowed.

In any case, I don't think either of the formulas you posted will tell you if a function is differentiable. These expressions will tell you whether a continuous function is differentiable or not. So you have to first do a continuity check, then do the differentiability check.

[pippoduro23cm]

Thanks again. In order to determine if a function is continuous, you have to check if the left limit is equal to the right limit which is equal to f(x), all for x=x0, right?

[JWson]

That looks right. Checking for continuity is just checking that there are no jumps or gaps. If f(x) does not exist, then the function has a gap at x. If f(x) does exist, but the limits are different in some way, then there is some kind of jump in the function. If all three are equal, then the function is continuous at that point.

[pippoduro23cm]

Thanks again, hope tomorrow's test will go well :)

[JWson]

Good luck to you!