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

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.

• – Mauro ALLEGRANZA Apr 6 '18 at 9:32
• Hello @MauroALLEGRANZA. Is this definition suitable for an analysis course? – Rabih Assaf Apr 6 '18 at 10:05
• See e.g William Ziemer, Modern real analysis, Springer (2nd ed 2017), page 36. – Mauro ALLEGRANZA Apr 6 '18 at 10:12
• It there a similar definition for uniform continuity using neighborhood? – Rabih Assaf Apr 6 '18 at 11:05

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

• If D\subset \mathbb{R}, then this distance at left would be the absolute value? – Rabih Assaf Apr 6 '18 at 11:06

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

• Hello, are these definitions considered as topological definitions? – Rabih Assaf Apr 6 '18 at 10:05
• Yes, in standard topology of real line these are topological definitions. If you use a different metric then the absolute values change to d(x,y). – Mohammad Riazi-Kermani Apr 6 '18 at 10:08
• @RabihAssaf they are the standard "metric definitions", we could replace the open balls in the domain by relatively open neighbourhoods, if you prefer. Uniform continuity needs a metric (or really, a uniformity) on the domain. There is no purely topological alternative, of course. – Henno Brandsma Apr 7 '18 at 6:16