Question

Show that the function f given by
f(x)=1/[x+2]^3 ∀x∈]-2,2[
is continuous but not bounded in ]− 2, 2 [

Answer 1

Answer on Question #69105 – Math – Real Analysis

Question

Show that the function $f$ given by $f(x) = \frac{1}{(x + 2)^3} \forall x \in (-2, 2)$ is continuous but not bounded in $(-2, 2)$.

Proof

Since the function $f(x) = \frac{1}{(x + 2)^3}$ is defined at any point from $(-2, 2)$, and $f$ is elementary, function then $f(x) = \frac{1}{(x + 2)^3}$ is continuous in $(-2, 2)$.

(see https://www.math24.net/continuity-functions/).

But since $\lim_{x\to -2 + 0}\frac{1}{(x + 2)^3} = \infty$ then $f$ is unbounded in $(-2, 2)$.

(see https://en.wikipedia.org/wiki/Bounded_function). In other words,

$\forall M > \frac{1}{64} \exists x_0 \in (-2, 2) \colon f(x_0) = M$. Solving the equation $\frac{1}{(x_0 + 2)^3} = M$ we get: $x_0 = \frac{1}{\sqrt[3]{M}} - 2$, so $f$ is unbounded in $(-2, 2)$.

The statement is completely proved.

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