What Defines Function Equality?

[zenctheo]

Hello to every one!
I have a question that came up when I was talking with a fellow mathematician.
I used to say that two functions are equal when the have the same formula and the same domain and codomain.
We read in a book though that two functions are equal when they have the same domain and when the values of the function are equal for the same X.
For example
$$f(x)=x^2$$ and $$g(x)=x^3$$ are equal when their domain is only the points 0 and 1,$$x \in \{0,1\}$$because f(0)=g(0)=0 and f(1)=g(1) even though their formula is different.
I thought that this definition of equality is incomplete because by saying that f(x)=g(x) then

$$\frac{df}{dx}=\frac{dg}{dx}$$ but on point x=1 $$\frac{df}{dx}=2$$ and $$\frac{dg}{dx}=3$$.

Thus we derive two different results from to equal quantities. Therefore two functions in order to be equal should also have the same formula.
Can you please give any insight on this?
Thanks a lot in advance.
Akis

 

[Ben Niehoff]

The derivative is not defined on the domain given. It requires a continuous interval. Remember the limit definition of the derivative:

$$f'(x) = \lim_{\Delta x \rightarrow 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$$

But for nearly all $\Delta x$, $x + \Delta x$ lies outside your domain. Therefore, you can't take the limit. :)

So, you are correct: Two functions are equal if and only if they have the same domain and their values are equal at every point within the domain.

 

[zenctheo]

Thanks a lot for the reply.
You that I am wrong because I was the one saying that the functions should also have the same formula.
In order to get things straight: You mean that the above two functions are equal... or not?

 

[LukeD]

The functions are in fact equal. Also, as Ben said, those functions don't have derivatives because they're not defined on an open interval of the real numbers.

As another example, would you consider these to be the same function?
Let's say f and g are functions from the real numbers to the real numbers defined as

f(x) = x
g(x) = x when x^2 >= 0 and -x when x^2 < 0

Since the functions are only defined on the real numbers, there are no points where they'd differ.

On a related note: "Having the same formula" is not a well-defined concept. Most (almost all) functions cannot be written with a closed formula and many (as you've seen with the example you gave) have multiple formulas.

 

[zenctheo]

Ok. It's nice to learn a new thing. Even if I am proven wrong

Thanks a lot.