# Statement(s) > [!NOTE] Theorem (Equivalence of Continuous and Sequentially Continuous Real Functions) >A given real-valued function is [[Continuous Real Function#^c86347|sequentially continuous]] at $c$ if and only if its [[Continuous Real Function#^2b83c3|continuous]] at $c$. > [!NOTE] Theorem (Equivalence of Continuous and Sequentially Continuous Functions Between Euclidean Spaces) > A function $f:U \subset \mathbb{R}^n \to \mathbb{R}^k$ is [[Continuous maps|continuous]] at $p\in U$, iff it is [[Sequentially Continuous Functions|sequentially continuous]] at $p$. # Proof(s) ###### Proof of Equivalence of Continuous and Sequentially Continuous Real Functions $(\implies)$ Continuous implies sequential continuous: Fix $\epsilon >0$. WTS that there exists $N \in \mathbb{N}$ such that: $n \geq N \implies |f(x_{n}) -f(c)|< \epsilon $Since $f$ is continuous at $c$, given any $\epsilon>0$ there exists $\delta >0$ such that $x \in E \text{ and } |x-c| < \delta \implies |f(x) - f(c)| < \epsilon \quad (1)$Now, since $x_{n} \to c$ there exists an $N \in \mathbb{N}$ such that $n \geq N \implies |x_{n} - c| < \delta \quad (2)$Therefore for $n \geq N$ it follows from (1) and (2) that $n \geq N \implies|x_{n} -c| < \delta \implies |f(x_{n}) - f(c)| < \epsilon$and so $f(x_{n}) \to f(c)$ as $n \to \infty$ as claimed. $(\impliedby)$ Conversely, we prove the contrapositive statement which is that: if $f$ is [[Discontinuous Function (Epsilon-Delta Definition)|discontinuous]] at $c$ then it is not sequentially continuous at $c$. If $f$ is discontinuous at $c$, then for each $n \in \mathbb{N}$ we take $\delta = \frac{1}{n}$ in the definition of discontinuity we can choose an $x_{n} \in E$ with $|x_{n} - c| < \frac{1}{n} \implies |f(x)-f(c)| \geq \epsilon$In this way we have constructed a sequence $(x_{n}) \in E$ such that $x_{n} \to c$ but $f(x_{n}) \not \to f(c)$ so $f$ is not sequentially continuous at $c$. ###### Proof of Equivalence of Continuous and Sequentially Continuous Functions Between Euclidean Spaces Analogous to above proof.