MATHEMATICAL ANALYSIS 04
Continuity
A continuous function preserves limiting behaviour: nearby inputs produce nearby outputs.
Continuity at a point
\[\boxed{\lim_{x\to a}f(x)=f(a)}\]
This requires the function value to exist, the limit to exist, and the two to agree.
Epsilon-delta form
For every \(\varepsilon\gt0\), there is \(\delta\gt0\) such that
\[|x-a|\lt\delta\Longrightarrow |f(x)-f(a)|\lt\varepsilon.\]Continuous on an interval
A function is continuous on an interval when it is continuous at every relevant point. Polynomials are continuous everywhere; rational functions are continuous wherever their denominators are nonzero.
Differentiability implies continuity
If \(f\) is differentiable at \(a\), then it is continuous at \(a\). The converse is false: \(|x|\) is continuous at \(0\) but not differentiable there.
Why continuity matters
The next major theorems require continuity. It prevents arbitrary jumps and allows global conclusions from local behaviour.