- #31
Oxymoron
- 868
- 0
Thankyou Matt. Using closed sets the continuity is easier to prove because of the topology.
Is the Mapping f:\mathbb{Q}_p \rightarrow \mathbb{R} Continuous?
- Thread starter Oxymoron
- Start date
-
- Tags
- Continuous
Click For Summary
SUMMARY
The mapping f:ℚₚ → ℝ, defined as f(x) = x, is not continuous. The proof relies on the definition of continuity in metric spaces, where the p-adic metric d(x,y) diverges as x approaches y in the Euclidean metric. Specifically, for any ε > 0, it is impossible to find a δ > 0 such that d(x,y) < δ implies d(f(x),f(y)) < ε, as demonstrated by the example where y = 2 leads to |0-2|ₚ < δ but |0-2| > 1. Thus, the function fails to meet the criteria for continuity.
PREREQUISITES- Understanding of metric spaces and continuity definitions
- Familiarity with p-adic numbers and their properties
- Knowledge of Euclidean metrics and their comparison to p-adic metrics
- Basic concepts of topology, particularly open and closed sets
- Study the properties of p-adic metrics in detail
- Explore the concept of continuity in various topological spaces
- Investigate the implications of embedding p-adic numbers into real numbers
- Learn about continuous functions and their characteristics in metric spaces
Mathematicians, particularly those focused on number theory and topology, as well as students studying advanced calculus and analysis concepts related to continuity and metric spaces.
Similar threads
- · Replies 3 ·
- · Replies 0 ·
- · Replies 2 ·
- · Replies 1 ·
- · Replies 3 ·
- · Replies 20 ·
- · Replies 6 ·
- · Replies 11 ·