# Thread: question on definition of continuity

1. ## question on definition of continuity

if X is a function from metric space X to Y and x is an element of X, i need to show that the definition of continuity given by (for each open neighborhood V of f(x), there exists an open neighborhood U of x s.t. f(U) is a proper subset of V) coincides with the definition given by (whenever {X_n} is a sequence in X s.t. x_n converges to x, then f(x_n) converges to f(x)).

i can't seem to relate the two definitions too well. help?

2. Originally Posted by squarerootof2
if X is a function from metric space X to Y and x is an element of X, i need to show that the definition of continuity given by (for each open neighborhood V of f(x), there exists an open neighborhood U of x s.t. f(U) is a proper subset of V) coincides with the definition given by (whenever {X_n} is a sequence in X s.t. x_n converges to x, then f(x_n) converges to f(x)).

i can't seem to relate the two definitions too well. help?
Let $\varepsilon>0$ and suppose for every open neighborhood $V$of $f(x)$, there exists an open neighborhood $U$ of $x$ such that $f(U) \subset V$.

Let $\left\{x_n\right\}$ be a sequence in $X$ such that $x_n \rightarrow x$. Then there is an $N \in \mathbb{N}$ such that if $n > N$, $x_n \in U$ implies $f(x_n) \in f(U) \subset V$ meaning $f(x_n) \rightarrow f(x)$.

for the other direction, try to prove it by contradiction.. Ü