Functions and Transcendental Numbers

[Agent Smith]

Hello there,

A function's continuity is not affected by the presence of transcendental numbers in an interval. Here’s why:

Key Points:

1. Continuity Definition:

• A function f(x) is continuous at a point a if lim(x -> a) f(x) = f(a). This holds regardless of the types of numbers.

2. Transcendental Numbers:

• These are just numbers that aren’t roots of any polynomial with rational coefficients. Examples include π and e.

3. Dense Subsets:

• Both rational and transcendental numbers are densely packed in the real numbers, but this doesn’t affect continuity.

Conclusion:

Continuity depends on how the function behaves as it approaches any point, not on the specific type of numbers. Therefore, a function can be continuous on an interval despite the presence of transcendental numbers.

Hope this helps!

Most helpful!

I conjecture that there are some numbers that are not solutions to any and all functions except perhaps trivially, as in in \(\displaystyle x - \pi = 0\). My first mathematical conjecture. Perhaps I can rephrase that as there are numbers that are not in any solution set.

So a function is continuous at a point \(\displaystyle a\) IFF \(\displaystyle \displaystyle \lim_{x \to a^{\pm}} f(x) = f(a)\). Correct?