how to understand the definition of continuity in analysis?

Question

Please have a look at the picture above. This is about the continuity in analysis. I don't really understand how to utilize this definition?

It says that is statement is equivalent to f is continuous at x. How to relate it to neighbourbood and open ball? Can anyone please help me out?

Answer 1

Hint:

This definition simply states that $f$ is continuos at $x$ iff $$\lim_{z\rightarrow x}\, f(z) = f(x)$$

It can be proved that it is equivalent to the topological definition of coninuity i.e. $f$ is continuous iff the image of an open subset of $\mathbb{R}$ through $f$ is an open subset of the set $\mathrm{Range}(f)$.

An open ball is a particular kind of neighbourbood, since $\forall r >0$, defining $B(x,r)$ you can always find an open ball, like $B(x,r/2)$ which is an open set completely inside $B(x,r)$.

Answer 2

The italicized statement is a definition of "$f$ is continuous at $x$" where $f$ is a function from a metric space $X$ to a metric space $Y$. Any neighbourhood of $f(x)$ in $Y$ contains an open ball $B_\epsilon(f(x)) = \{y: d_Y(y,f(x))<\epsilon\}$. The statement says that for every such neighbourhood of $f(x)$ there is an open ball $B_\delta(x) = \{z: d_X(z,x) < \delta\}$ that $f$ maps into the neighbourhood.