Continuity and Uniform Continuity for functions of one variable (topological definitions)in analysis course

Question

Hello i need to know what is the exact meaning of this expression: Continuity and Uniform Continuity for functions of one variable (topological definitions) in analysis course. Does it mean the usual definition:

If $f:D\rightarrow \mathbb{R}$ is a function and $a\in D$ then we say that f is continuous at a if: $(\forall \epsilon>0) (\exists \delta>0) (\forall x\in D) , |x-a|< \delta \rightarrow |f(x)-f(a)|<\epsilon$.

Or it's the same definition but using metric distances instead of absolute values.

Answer 1

You should use distance on the left and absolute value on the right, that is,$$d(x,a)<\delta\implies\bigl|f(x)-f(a)\bigr|<\varepsilon.$$

Answer 2

The definition of continuity at a point is

If $$f:D→R$$ is a function and $a∈D$ then we say that $f$ is continuous at $a$ if: $$(∀ϵ>0)(∃δ>0)(∀x∈D),d(x,y)<δ \implies |f(x)−f(a)|<\epsilon .$$

A function is continuous on $D$ if it is continuous at every point $x\in D$

On the other hand the uniform continuity is defined as

If $$f:D→R$$ is a function, then we say that $f$ is uniformly continuous on $D$

if: $$(∀ϵ>0)(∃δ>0)(∀x , y ∈D),d(x,y)<δ\implies |f(x)−f(y)|<\epsilon .$$