Show that the real valued identity function is continuous How do I show with the definition of continuity that $f : \mathbb{R} \to\mathbb{ R}$ defined by $f(x)= x$ for all $x\in\mathbb{R}$ is continuous?
I understand that this function is continuous, however I am having a hard time proving it. Especially because we have been using converging sequences to prove the continuity of functions and I am not sure if the book wants met to do that here as well.
If so I think it would be something like this:

Pick an $(x_k)_{k∈\mathbb{N}}$ in $A$ that converges to $a$.
Pick an $ε > 0$ , then we can find an $δ > 0$ such that $||f(x)-f(a)|| < ε$ for all $x ∈ A$ with $||x-a|| < δ$.
Because $(x_k))_{k∈\mathbb{N}}$ converges to $a$. We can find an $k_0 ∈ \mathbb{N}$ such that $||x_k-a|| < δ$ for all $k\ge k_0$. Pick a $k\ge k_0$ then $||f(x_k)-f(a)|| < ε$ and thus $(f(x_k))_{k ∈ \mathbb{N}}$ converges to $f(a)$.

Would this be correct. Maybe someone could show me how to prove this without using converging sequences.
 A: Let $f$ be the identity function $\mathbb{R} \to \mathbb{R}$. Here are six weird reasons why $f$ is continuous (number 2 will make you cry, then number 3 will restore your faith in humanity!)


*

*The identity trivially satisfies that the preimage of any open set is open.

*For every $\epsilon$, pick $\delta = \epsilon$. Then $|x-y| < \delta$ implies $|f(x) - f(y)| < \epsilon$. So the identity is in fact uniformly continuous.

*$f$ is differentiable: its derivative is $\lim_{h \to 0} \frac{x+h-x}{h} = 1$. Therefore it is continuous.

*If $(x_n) \to x$, then $(f(x_n)) = (x_n) \to x = f(x)$, so $f$ preserves the convergence of sequences.

*Consider the natural hyperreal extension which I will write $\bar{f}$ because my MathJax-foo is not good enough and I don't have access to \prescript. Then if $dx$ is infinitesimal, we have $\bar{f}(x+dx) - \bar{f}(x) = x + dx - x = dx$, which is infinitesimal.

*$f$ is monotone and surjective. (OK, I don't know of a way of proving this that doesn't basically boil down to the epsilon-delta of method 2. But it's nice to know.)
