# Pedagogically weaker notions of continuity

We introduce a weaker notion of continuity, similar to continuity at a point, that might be useful pedalogically.

## Continuity

Over \(\mathbb{R}\), a function \(f: \mathbb{R} \to \mathbb{R}\) is defined to be continuous if

\[\forall x \, \forall \epsilon \, \exists \delta\, \forall y \text{ s.t. } \vert y-x\vert < \delta \implies \vert f(y)-f(x)\vert < \epsilon\]where \(x, y \in \mathbb{R}, \epsilon, \delta \in (0, \infty)\)

Letting \(R\) be the statement that \(\forall y \text{ s.t. } \vert y-x\vert < \delta \implies \vert f(y)-f(x)\vert < \epsilon\), we could see \(f\) is continuous if and only if

\[\forall x \, \forall \epsilon \, \exists \delta\, R\]## Weaker notions of continuity

The most common weaker notion of continuity is that of continuity at a point. We say that \(f\) is continuous at \(x\) if

\[\forall \epsilon \, \exists \delta\, R\]and as such it’s apparent that \(f\) is continuous if and only if it is continuous at all \(x\). Inspired by this, we define that \(f\) is \(\epsilon\)-continuous for some \(\epsilon>0\) if

\[\forall x \, \exists \delta R\]**Example:** The floor function is \(2\)-continuous but not \(\epsilon\)-continuous for any \(\epsilon \leq 1\).

**Example:** If a function is bounded then it is \(\epsilon\)-continuous for some \(\epsilon\).

It is clear that \(f\) is continuous if and only if it is continuous at all \(\epsilon\). However, \(\epsilon\)-continuity has rather unusual properties.

**Claim:** If \(f\) is \(\epsilon_1\)-continuous and \(g\) is \(\epsilon_2\)-continuous then \(f+g\) is \(\epsilon_1 + \epsilon_2\)-continuous.

**Example:** The exponential function is continuous but \(\exp(x)\text{floor}(x)\) is not \(\epsilon\)-continuous for any \(\epsilon\).

**Example:** The exponential function is continuous but \(\exp(\text{floor}(x))\) is not \(\epsilon\)-continuous for any \(\epsilon\).

Much of the ideas driving the above claims and examples could be traced back to the proofs of preservation of continuity under addition and multiplication.

## Uniform continuity

Similarly we can say that \(f\) is uniformly \(\epsilon\)-continuous if

\[\exists \,\delta\, \forall x R\]Again \(f\) is uniformly continuous if and only if it is uniformly \(\epsilon\)-continuous for all \(\epsilon\). Uniform \(\epsilon\)-continuity also clearly implies \(\epsilon\)-continuity.

**Example:** \(\text{floor}(\exp(x))\) is 2-continuous but not uniformly 2-continuous.

## Acknowledgements

Thanks to Isaac Li for discussions about this topic, especially on the use of floor and exponential functions to generate counterexamples.