Continuity of the function $\mathbb{R}^k \to\mathbb{R}: x\mapsto \ln(1+ \lVert x \rVert)$

Question

Examine the continuity of the function $f\colon\mathbb{R}^k \to \mathbb{R}$ defined by $f(x) = \ln (1+ \lVert x \rVert)$, where $\lVert\cdot\rVert$ is a norm.

Answer 1

Let $c \in \mathbb{R}$; if $x \in \mathbb{R}$, then $|f(x) - f(c)| < \varepsilon$ iff $\log ( 1+ |c|) - \varepsilon < \log(1+|x|) < \log(1+ |c|) + \varepsilon$, which holds iff $$ \frac{1+|c|}{e^{\varepsilon}} - 1 - |c| < |x| - |c| < (1+|c|)e^{\varepsilon} - 1- |c|; $$ this is implied by $$ \big| |x| - |c| \big| < \min \bigg\{ \bigg| \frac{1+|c|}{e^{\varepsilon}} - 1 - |c| \bigg|, (1 + |c|)e^{\varepsilon} - 1 - |c| \bigg\} =: \delta; $$ so if $|x-c| < \delta$, then $| |x| - |c|| \leq |x-c| < \delta$, whence $|f(x) - f(c)| < \varepsilon$.

Answer 2

It's the composition of continuous functions $x\mapsto \|x\|$, $x\mapsto x+1$, $x\mapsto \ln x$.