Is $ f(x) = \frac{x}{x}$ differentiable? I know that the function $f(x) = \frac{x}{x}$ is not differentiable at $x = 0$, but according to the definition of differentiable functions:

A differentiable function of one real variable is a function whose derivative exists at each point in its domain

since $x = 0$ is not in the domain of $f$, it doesn't have to be differentiable at that point for the function to be differentiable. This suggests that $f$ is differentiable as every other points in the domain has a derivative of $0$.
However, some say that a function must be continuous if it's differentiable. This disproves the fact that $f$ is differentiable since it's not a continuous function.
Then is it really a differentiable function?
 A: $\frac xx$ is continuous at all points in its domain.  It has derivative $0$ at all points of $\Bbb R$ except $0$.
A: 
"I know that the function $f(x)=\frac xx$ is not differentiable at $x=0$"

The function isn't ANYTHING at $x=0$.  $0$ is not in the domain.

since x=0 is not in the domain of f, it doesn't have to be differentiable at that point for the function to be differentiable. This suggests that f is differentiable as every other points in the domain has a derivative of 0.

Absolutely correct.

However, some say that a function must be continuous if it's differentiable. This disproves the fact that f is differentiable since it's not a continuous function.

But $f$ is a continuous function.  It's continuous at every point of its domain.... which doesn't include $0$.
A: The function $f(x) = \frac{x}{x}$ has a natural domain of $\Bbb{R} \setminus \{0\}$; you can sensibly substitute in any value except $0$. In order to be continuous or differentiable at a point, it is required that the point be part of the domain of the function. The function $f$ is therefore neither continuous nor differentiable at $0$.
Does this mean the function is continuous? It depends on who's asking. Some people define the term "continuous" as being defined and continuous at every point on $\Bbb{R}$, and say that all points where the function is undefined are points of discontinuity by default. Others say that a function is "continuous" if it is continuous at every point on its domain. In the latter sense, $f$ is continuous and differentiable. In the former sense, $f$ is neither.
A: To see why the function $f(x) = \frac{x}{x}$ is differentiable everywhere except at $0$, and has derivative equal to $0$ where it is differentiable, consider the following: 
The graph of $f(x) = \frac{x}{x}$ is the graph of $y=1$ with the point $(0,1)$ removed. Hence, $f'(x) = 0$ over $(-\infty, 0) \cup (0, \infty)$  and $f(x)$ is not differentiable at $x=0$. 
