Determine if the function $f(x)$ is integrable in $[0,1]$

$$\displaystyle f( x) =\begin{cases}
\frac{1}{x^{2}} & x >0\\
0 & x=0
\end{cases}$$

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***Attempt:***

Let's check the limits of each side $0^+$ to show that $\frac{1}{x^2}$ is not bounded.

$$\displaystyle \lim \_{x\rightarrow 0^{+}}\frac{1}{x^{2}} =\infty $$

Therefore, the function is not bounded when $\lim\_{x\rightarrow 0^{+}}$
, and is not integrable in $[0,1]$

In addition, there is a well-known theorem:

> 
> If $f$ is a continuous function except for the finite number of unconsciously points $\Rightarrow$ $f$ is integrable.
> 
> 
> 

Which contradicts the attempt.