Let $f$ and $g$ be continuous functions defined on an open interval $I$. Let $a\in I$ such that $f(a)<g(a)$. Show that there exists an open interval $J\subset I$ with $a\in J$ such that $f(x)<g(x)$ for all $x\in J$
I'm honestly struggling to even conceptualize what this problem is asking me to prove let alone to how start proving it. Can anyone guide me or help me through?