The assignment displayed in your screenshot contains a problem from Spivak's Calculus (Problem 8.8) focused on proving properties of a monotonically increasing function $f$ defined on an open interval $I = (a, b)$. Here’s a breakdown of the problem requirements:

Problem Statement Analysis

1. Monotonicity: We assume $f$ is monotonically increasing, meaning $f(x) \leq f(y)$ for all $x < y$ within the interval $I$.
2. Jump Discontinuity Definition: For any $c \in I$, the jump of $f$ at $c$ is given by: $[f(c)] := \lim_{x \to c^+} f(x) - \lim_{x \to c^-} f(x)$

Required Proofs

• Part (a): Show that the one-sided limits $\lim_{x \to c^+} f(x)$ and $\lim_{x \to c^-} f(x)$ exist for each $c \in I$.
• Part (b): Prove that $[f(c)] \geq 0$ and that $[f(c)] = 0$ if and only if $f$ is continuous at $c$.

Strategy for Solutions

1. One-Sided Limits: Since $f$ is monotonically increasing, it implies that the left and right limits as $x$ approaches $c$ should exist due to the bounded behavior of monotonically increasing functions on intervals.

2. Jump Calculation:

   • If there’s a discontinuity at $c$, the jump $[f(c)]$ captures this gap.
   • Since $f$ is increasing, the jump $[f(c)]$ should be non-negative because $f(x)$ does not decrease on $I$.

3. Continuity Condition:

   • If $f$ is continuous at $c$, then $\lim_{x \to c^+} f(x) = \lim_{x \to c^-} f(x) = f(c)$, leading to $[f(c)] = 0$.
   • Conversely, if $[f(c)] = 0$, then $f$ must be continuous at $c$ since there’s no gap between the left and right limits.

Would you like a detailed solution with formal proofs for each part?

Follow-Up Questions

1. What is the general impact of a function's monotonicity on the existence of one-sided limits?
2. How does the concept of jump discontinuity relate to the continuity of functions?
3. Can the result $[f(c)] \geq 0$ be generalized to all monotonically increasing functions?
4. What would happen if $f$ were monotonically decreasing instead?
5. How do these properties change if the function is not defined on an open interval?

Tip

In problems involving continuity and limits, always start by examining the behavior of the function as it approaches points from both directions. This helps in understanding discontinuities and ensuring proper limit evaluation.