The problem asks us to analyze a function $f$ that is defined on an open interval $I = (a, b)$ and is monotonically increasing. Specifically:

Given:

1. $f(x) \leq f(y)$ for all $x < y$ in $I$.
2. For any point $c \in I$, define the jump of $f$ at $c$ as: $[f(c)] := \lim_{x \to c^+} f(x) - \lim_{x \to c^-} f(x).$

Tasks:

1. (a) Prove that the one-sided limits $\lim_{x \to c^+} f(x)$ and $\lim_{x \to c^-} f(x)$ exist for all $c \in I$.
2. (b) Prove that $[f(c)] \geq 0$. Also, show that $[f(c)] = 0$ if and only if $f$ is continuous at $c$.

Solution Outline:

Part (a) - Existence of One-Sided Limits

1. Since $f$ is monotonically increasing, it’s bounded on either side of $c$ within $I$.
2. A theorem on monotonic functions states that they have finite one-sided limits at every point within their domain.
3. Therefore, $\lim_{x \to c^+} f(x)$ and $\lim_{x \to c^-} f(x)$ must exist.

Part (b) - Non-Negativity of the Jump

1. By definition of a monotonic function, $\lim_{x \to c^+} f(x) \geq \lim_{x \to c^-} f(x)$.
2. Thus, $[f(c)] \geq 0$.
3. If $[f(c)] = 0$, it implies $\lim_{x \to c^+} f(x) = \lim_{x \to c^-} f(x)$, meaning $f$ is continuous at $c$.

Would you like a detailed step-by-step solution for each part?

Here are some related questions to deepen understanding:

1. What is a monotonic function, and how does it affect the limits?
2. Why do monotonic functions have one-sided limits at every point in their domain?
3. How does the definition of continuity relate to the jump function in this context?
4. What would happen to the jump if $f$ were decreasing instead?
5. Can a function have discontinuities but still be monotonic?

Tip: For any monotonic function, knowing that one-sided limits exist is often useful for analyzing its behavior at critical points.