Sum of non uniformly continuous and uniformly continuous is non uniformly continuous?

Question

I have this question:

Prove/disprove: $f(x)=x\ln x+\sin(x^3)\frac1x$ is uniformly continuous in $(0,\infty)$.

I know that $\sin(x^3)\frac1x$ is uniformly continuous and $x\ln x$ is non uniformly continuous. I tried proving that the sum of uniformly continuous function and non uniformly continuous function is non uniformly continuous, but is this even true?

Answer 1

The sum of two uniform continuous maps $f,g$ is uniformly continuous.

If $f$ is uniformly continuous and $g$ is not, $f+g$ can't be uniformly continuous as otherwise $g = (f+g) - f$ would be uniformly continuous.