A one-sided discrete intermediate value theorem

In this post the author states a discrete version of the Intermediate Value Theorem as follows:

For integers $ a < b $, let $ f $ be a function from the integers in $ [a, b] $ to $ \mathbb{Z} $ that satisfies the property, $ |f(i + 1) - f(i)| \leq 1 $ for all $ i $. If $ f(a) < 0 < f(b) $, then there exists an integer $ c \in (a, b) $ such that $ f(c) = 0 $.

In this post a one-sided discrete intermediate value theorem was proposed:

Let $f$ be an integer-valued function defined on the integers in $[a, b]$.

(1) If $ f(i + 1) - f(i) \geq -1 $ for all $ a≤i≤b-1 $ and $ f(b) < 0 < f(a) $, then there exists an integer $ c \in (a, b) $ such that $ f(c) = 0 $.

(2) If $ f(i + 1) - f(i) \leq 1 $ for all $ a≤i≤b-1 $ and $ f(a) < 0 < f(b) $, then there exists an integer $ c \in (a, b) $ such that $ f(c) = 0 $.

Then my question is about the validity of this result since I am not confortable with it.