**Theorem** The function $f: \mathbb{R} \to \mathbb{R}$ given by $f(x)=\lambda$, $\lambda \in \mathbb{R}$ is a constant, is [[Continuous Real Function|continuous]] at every $c \in \mathbb{R}$. **Proof (using [[Continuous Function (Epsilon-Delta Definition)|epsilon-delta]])** Fix $c \in \mathbb{R}$, then for any $x\in \mathbb{R}$ $|f(x)-f(c)| = 0 < \lambda$for any $\epsilon, \lambda >0$ such that $|x-c|<\epsilon$. This 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 $f(x_{n}) = \lambda \to \lambda = f(c)$ as $n \to \infty$. So $f$ is continuous.