Why would the function $f(x)= \frac{1}{x^5(\exp(\frac{1}{x^5})-1)}$ not be continous?

Question

I was graphing the function $$f(x)=\frac{1}{x^5(\exp(\frac{1}{x^5})-1)}$$ and I noticed that somewhere around $x=1124.925$ and $x=1124.926$ the function stopped being continuous. I was initially graphing it with Desmos but repeating it with other graphing software (GeoGebra) and the calculator in my phone, (FSC) since I was finding an error of the software easier to believe, yielded the same results. Right around that same value, all of those jumped from $f_{(1124.925)}\approx 0.83$ to $f_{(1124.926)}\approx 1.25$. Now I have no idea if there is some kind of problem in my maths or if it has to do with how computers calculate things. If there is no problem in the software, why is this happening, and why is this function not continuous?

(Here is the Desmos link)

Answer 1

Assuming the denominator is never zero and x is never zero, as a composition of continuous functions your function is continuous as well. In fact, it is even differentiable on R_>0 with the same reasoning.

The plotting software you're using approximates the function values; it doesn't evaluate for every (more than countable) value of x, and it only uses machine precision, which means there are at least two possible sources for the errors perceived in the plot (as well as the possibility that the actual plotting is faulty).

Answer 2

Almost all have been said.

Compute for $x=1000$; the function value is $$0.9999999999999995000000000000000833333333333333333$$ For large values of $x$, use instead $$f(x)\sim1-\frac{1}{2 x^5}+\frac{1}{12 x^{10}}+O\left(\frac{1}{x^{20}}\right)$$ which will give the same result.