I encountered fixed points in various theories like in case of real analysis while studying about fixed points, suppose $f(x)$ is a real-valued function then fixed points of $f$ are given by $x$ satisfying $f(x) = x$.
While considering Dynamical systems for 'flows' where say for one-dimensional case, the governing equation is $\dot{x} = f(x)$, in that case the fixed points are given by $\dot{x} = 0$ or $f(x) = 0$.
Are the two definitions equivalent in some way? If so how are they related?
They are related as follows.
Let $x(0)=a$, and for nonnegative integers $n$, let $x(n+1)=f(x(n))$.
(Sometimes people write this as $x(n)=f^{(n)}(a)$.)
Then $f(a)=a$ implies that $x(n)$ is constant.
Analogously, if $x(t)$ is a function of a real variable $t$, then $\frac{dx}{dt}=\dot{x}=0$ iff $x(t)$ is constant.