Differentiability of a singleton set

Question

Let $f$ be a function $ f:\{2\} \rightarrow\mathbb R$. Now absolutely $f$ is continuous. But what we can say about the differentiability of the function $f$ at $x=2$ ?

Answer 1

The function $f$ is not differentiable at $2$ since $2$ is not an accumulation point of the domain of $f$.

Answer 2

By definition, a function is differentiable at $x=2$ if the limit $\lim_{x\to\ 2}\frac{f(x)-f(2)}{x-2}$ exists. But in this case, since $2$ is the only value of $x$ for which $f(x)$ is defined, and $x-2=0$ for that value of $x$, the expression $\frac{f(x)-f(2)}{x-2}$ is not defined for any $x$. Therefore, the limit cannot exist.

Answer 3

From the viewpoint of smooth manifold, you treat $\{2\}$ as a $0$-dimensional manifold. Any function on a $0$-dimensional manifold is automatically differentiable. But the derivative would not be a slope of some tangent line, but rather the uninteresting zero linear map $T:\{\vec0\}\to\Bbb R$, $T(\vec0)=0$.