On Bartle's 'Elements of Integration' Exercise 7.T

Question

My question is twofold:

1. Are there errors in my proof?
2. Are there more elegant concise ways to do it? In particular I was hoping to find a way for the converse part to work without having to resort to a discrete measure space by using a function of the form $f(x) = x^{\frac{\alpha - 1} p}, 1 > \alpha > 0$ on $[0, 1]$ with $\alpha$ chosen carefully. I believe that should be possible but I don't know how.

Problem statement (R.G. Bartle 'The Elements of Integration', p. 79 Exercise 7.T):

Let $(X, \mathbf X, \mu)$ be a finite measure space and let $1 \leq p < \infty$. Let $\varphi$ be continuous on $\mathbb R$ to $\mathbb R$ and satisfy the condition: $(*)$ there exists $K > 0$ such that $|\varphi(t)| \leq K |t|$ for $|t| \geq K$. Show that $\varphi \circ f$ belongs to $L_p$ for each $f \in L_p$. Conversely, if $\varphi$ does not satisfy $(*)$, then there is a function $f$ on $L_p$ on a finite measure space such that $\varphi \circ f$ does not belong to $L_p$.

Proof:

$\Longrightarrow$

Let $M = \max_{t \in [0, K]} |\varphi(t)|^p$; ($M < \infty$ as $[0, K]$ compact and $\varphi$ continuous)
$(*) \Rightarrow$ $|\varphi \circ f|^p = |\varphi \circ f|^p (\chi_{|f|^{-1}([0, K[)} + \chi_{|f|^{-1}([K, \infty[)}) \leq M \chi_{|f|^{-1}([0, K[)} + K^p |f|^p \chi_{|f|^{-1}([K, \infty[)} \leq M + K^p |f|^p;$
$\Rightarrow$ $||\varphi \circ f||_p^p = \int |\varphi \circ f|^p \mathrm d \mu \leq \int M + K^p |f|^p \mathrm{d} \mu = M \mu(X) + K^p ||f||_p^p < \infty$. (finite measure space and $f \in L_p$) $(\Rightarrow \varphi \circ f \in L_p)$

$\Longleftarrow$

$\lnot (*) \Rightarrow \forall K > 0 \; \exists |t| \geq K: |\varphi(t)| > K |t|$;
Let $t_n \in \mathbb R, t_n > n, |\varphi(t_n)| > n |t_n|, n \in \mathbb N$;
Let $(X = \mathbb N, 2^{\mathbb N}, \mu)$ be a measure space with measure $\mu(E) = \sum_{n \in E} \frac 1 {{|t_n|}^p n^2}$;
$\Rightarrow \mu(X) = \sum^{\infty}_{n = 1} \frac 1 {{|t_n|}^p n^2} \leq \sum^{\infty}_{n = 1} \frac 1 {n^2} < \infty$; ($\Rightarrow$ the measure space is finite)
Let $f:\mathbb N \ni n \mapsto t_n$;
$\Rightarrow ||f||_p^p = \int |f|^p \mathrm d \mu = \sum_{n=1}^{\infty} \frac{|f(n)|^p} {{|t_n|}^p n^2} = \sum_{n=1}^{\infty} \frac 1 {n^2} < \infty; (\Rightarrow f \in L_p)$ $\Rightarrow ||\varphi \circ f||_p^p = \int |\varphi \circ f|^p \mathrm d \mu = \sum_{n=1}^{\infty} \frac{|\varphi(f(n))|^p} {{|t_n|}^p n^2} = \sum_{n=1}^{\infty} \frac{|\varphi(t_n)|^p} {{|t_n|}^p n^2} \geq \sum_{n=1}^{\infty} \frac{1} {n^{(2-p)}} \geq \sum_{n=1}^{\infty} \frac{1} {n} = \infty.$
$(\Rightarrow \varphi \circ f \notin L_p)$

$\square$

I'm grateful for any suggestions for a continuous variant of the converse. Thanks in advance.