Notes - Continuous Mathematics HT23, Continuity

Continuity

@Define what it means for a function $f : D \to \mathbb{R}, D \subseteq \mathbb{R}$ to be continuous at a point $x$.

\[\lim _ {h\to 0} f(x+h) = f(x)\]

@Define what it means for a function $f : D \to \mathbb{R}, D \subseteq \mathbb{R}^n$ to be continuous at a point $\pmb x$.

\[\lim _ {\pmb{h}\to0} f(\pmb{x} + \pmb{h}) = f(\pmb{x})\]

Is the composition of two continuous functions continuous?

Yes.

Is $ \vert f \vert $ continuous if $f$ is?

Yes.

Under what conditions is $f^n$ continuous if $f$ is?

\[n \in \mathbb{N}\]

Under what conditions is $\log f$ continuous if $f$ is?

$f$ positive.

Under what conditions is $f/g$ continuous if $f$ and $g$ are?

$g$ nonzero.

@Define what it means for a function $f : D \to \mathbb{R}, D \subseteq \mathbb{R}$ to be differentiable at a point $x$.

\[d = \lim _ {h\to 0} \frac{f(x+h)-f(x)}{h} \text{ exists}\]

How can you view the operator $\frac{\text{d}\space}{\text{d}x}$ as a function?

It takes in a function and produces a new function.

Is the composition of two differentiable functions differentiable?

Yes.

Is the maximum of two differentiable functions differentiable?

No.

Is the modulus of a differentiable function differentiable?

No.