Answer
Continuous at $ x=a $
Work Step by Step
Recall that if $ f $ is a polynomial function, then we have $\lim_\limits{x\to a}f(x)=f(a)$.
$\lim_\limits{x \to a} f(x)= \lim_\limits{x \to a}(x^3+5x-1)=a^3+5a^2-1$
So, $\lim_\limits{x \to a} f(x)$ exists.
Now, $\lim_\limits{x \to a} f(x)=a^3+5a^2-1$
and $ f(a)=a^3+5a^2-1$
so, $\lim_\limits{x \to a} f(x) =f(a)$
Therefore, the function is continuous at $ x=a $
