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 $\displaystyle \varepsilon>0$ and suppose for every open neighborhood $\displaystyle V$of $\displaystyle f(x)$, there exists an open neighborhood $\displaystyle U$ of $\displaystyle x$ such that $\displaystyle f(U) \subset V$.

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

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