Question

How do I know whether or not a function is continuous?

Answer 1

A function is continuous at point p if the limit at x=p exists and\[\lim_{x \rightarrow p}f(x)=f(p)\]

Answer 2

...the limit exists iff\[\lim_{x \rightarrow p^-}f(x)=\lim_{x \rightarrow p^+}f(x)\]

Answer 3

a specific example would help

Answer 4

Let \[f:I \rightarrow \mathbb{R}.\] Then f is continuous at a point \[p \in I\] if \[\forall \varepsilon > 0 \exists \delta > 0 : \forall x \in I, |x-p|<\delta \Longrightarrow |f(x)-f(p)|<\varepsilon .\]

Notice that the value of delta actually depends on epsilon and p. So if the function if continuous on I, then at different points in the interval I you will need to select different values for delta if given some epsilon. A nice example is 1/x this function is continuous on (0, +infty) and if we take a delta-neighborhood of p really close to zero, we see that we have a large epsilon-neighborhood of f(p). If you picked p not so close to zero, then the delta-neighborhood of p would not be so large.

You might encounter something called uniform continuity, which is the case when convergence is everywhere the same, i.e. where delta does not depend on p, but on epsilon alone. That's all I can remember from functional analysis lol