# Definitions > [!Note] Definition (Continuity at a point) > Let $f: E \to \mathbb{R}$ be a [[Real Function|real function]], where $E \subset \mathbb{R}$. We say $f$ is *continuous* at $c \in E$ if for any $\epsilon >0$ there exists a $\delta >0$ such that $ x \in E \; \text{ and } \; |x-c| < \delta \implies |f(x) - f(c)| < \epsilon$ **Language**: If $f$ is continuous at every point in $E$ we say that $f$ is continuous on $E$. If $E= \mathbb{R}$ and $f$ is continuous at every point in $E$, then we just say that $f$ is continuous. > [!NOTE] Definition (Sequential continuity at a point) >The [[Real Function|real valued function]] $f:E \subset \mathbb{R} \to \mathbb{R}$ is *sequentially continuous* at $c \in E$ if whenever $(x_{n}) \in E$ and $x_{n} \to c$ as then $f(x_{n}) \to f(c)$ as $n \to \infty$. ^c86347 **Note**: By [[Equivalence of Continuous and Sequentially Continuous Functions Over Euclidean Spaces]] the above definitions are equivalent. > [!NOTE] Definition 3 > the [[Limit of Real Function at a Point|limit]] of $f$ at $c$ equals $f(c).$ # Properties Note [[Preservation of Inequalities by continuous real functions]] and [[Algebra of Continuous Real Functions]]. **Cts function on closed bounded interval**: Any real-valued function $f$ that is continuous a closed bounded interval is also [[Continuous Real Function on Closed Real Interval is Uniformly Continuous|uniformly continuous]] on the same interval and so it is [[Continuous real functions are Riemann Integrable|Riemann integrable]]. Other properties of $f$ are given by [[Intermediate Value Theorem (IVT)]] & [[Extreme Value Theorem]]. See [[Inverse of Strictly Monotonic Continuous Real Function Exists, is Strictly Monotonic in the Same Sense, and is Continuous]]. # Applications - [[Constant Function is Continuous]]. - [[Identity Function on Reals is Continuous]]. - [[Reciprocal Function is Continuous at non-zero points]]. - [[Thomae's function is discontinuous at rational points and continuous at irrational points]]. - [[Lipschitz Function]]. - [[Fréchet Differentiation]].