**Theorem** The function $f: \mathbb{R} \to \mathbb{R}$ given by $f(x) =x$ is [[Continuous Real Function|continuous]]. **Proof (using [[Continuous Function (Epsilon-Delta Definition)|epsilon-delta]])** Fix $c \in \mathbb{R}$, then for any $x\in \mathbb{R}$ $|f(x)-f(c)| = |x-c|$Given $\epsilon>0$, take $\delta = \epsilon$. Whenever $|x-c|< \delta$, we have $|f(x)-f(c)|= |x-c| < \epsilon$which shows that $f$ is continuous. **Proof (using [[Continuous Function (Sequential Continuity Definition)|sequence]])** Fix $c\in \mathbb{R}$. Take $x_{n} \in \mathbb{R}$ with $x_{n} \to c$ Then $f(x_{n}) = x_{n} \to c = f(c)$ as $n \to \infty$. So $f$ is continuous.