1
vote
1answer
31 views

Definition of a continuous function

I am struggling to understand a basic definition of a continuous function from a textbook: A function f is continuous if for all x, and for all $\epsilon>0$, there exists $\delta>0$ such that ...
0
votes
2answers
39 views

Countably or Uncountably Many Discontinuities

I want to know why the following function has uncountably many discontinuities: $$f(x)=\left\{\begin{array} & x^2 & x \not \in \mathbb{Q} \\ 0 & \text{otherwise} \end{array}\right .$$ ...
1
vote
1answer
39 views

What's behind the function $g(x)=\operatorname{inf}\{f(p)+d(x,p):p\in X\}$?

In several books on measure theory, I have seen the following problem: Suppose $(X,d)$ is a metric space, on which $f$ is a nonnegative lower semicontinuous function. Show that $f$ is the ...
2
votes
0answers
151 views

Intuition behind continuity in topological spaces

I was approaching the following problem: "Let $f \colon X \to Y$ be continuous. Is it true that if $x$ is a limit point of $A \subset X$ then $f(x)$ is a limit point of $f(A)$?" The answer is that ...
1
vote
1answer
58 views

$|f(x)|<|g(x)|$ and $\int g(x)<\infty\Rightarrow\ \int f(x)<\infty$

Let f,g continious functions on $[0,\infty)$ s.t $\forall x\in[0,\infty), |f(x)|\le|g(x)|$. Prove or give counterexample thtat if $\int_ 0^\infty g(x)dx<\infty$ then $\int_0^\infty ...
2
votes
2answers
70 views

Understanding Continuity of Functions

I know that graphically a function $f(x)$ is said to be continuous in $[a,b]$ if there are no breaks in the curve for $f(x)$ in the interval $[a,b]$ I also know that by definition, a function $f(x)$ ...
11
votes
8answers
6k views

Continuous versus differentiable

A function is "differentiable" if it has a derivative. A function is "continuous" if it has no sudden jumps in it. Until today, I thought these were merely two equivalent definitions of the same ...