It is well-known that if $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is a $C^1$ function with Lipschitz gradient, then $f$ satisfies the descent condition
$$f(y)\le f(x)+\left\langle\nabla f(x),y-x\right\rangle+\dfrac{L}{2}\left\|y-x\right\|^2,\text{ for all }x,y\in\mathbb{R}^n.$$
Results about this descent condition can be found in Xingyu Zhou's notes. It seems that the converse implication does not hold, i.e., there is some function with non-Lipschitzian gradient satisfying the descent condition. However, I tried functions $f(x)=x|x|$ and $f(x)=x^3$, but they do not work. Could you please give any help?
The answer is given by Xingyu Zhou in his notes with $f(x)=-e^x$ for all $x\in\mathbb{R}$.