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Conditional statements, also known as 'if-then' statements, play a critical role in mathematics. They form the basis of logical reasoning by positing that if one statement (the hypothesis) is true, then another statement (the conclusion) will also be true.

For instance, consider the statement: 'If a number is even, then it is divisible by 2.' This represents a straightforward conditional where the truth of the number being even (hypothesis) necessitates the truth of it being divisible by 2 (conclusion).

In the context of functions, as seen in our exercise, the conditional given is 'If a function is differentiable at a point, then it is continuous at that point.' To understand and test such statements, one must grasp the concepts of both differentiability and continuity. It's also essential to comprehend the contrapositive and negation of these conditionals. The contrapositive of the original conditional 'If not continuous, then not differentiable' has the same truth value, whereas the negation would be 'It is differentiable and not continuous,' contradicting the original assertion.

Understanding such logical connections is vital, as it assists in examining the interplay between different mathematical concepts.

Real-valued functions are functions that take a real number as input and produce a real number as output. This vast category of functions is a cornerstone of calculus and advanced mathematics.

The function described in the original exercise, where we discuss continuity and differentiability, is precisely this kind. A real-valued function might showcase a wide range of behaviors, and understanding these is key to mastering calculus in particular.

Continuity and differentiability are two fundamental properties of real-valued functions. A function is continuous at a point if there are no breaks, jumps, or holes at that point. Mathematically, a function is continuous at a point if the limit as you approach the point equals the function's value at the point.

Differentiability, on the other hand, refers to the existence of a function's derivative at a point, meaning the function's graph has a well-defined tangent at that point. A remarkable aspect of real-valued functions is that if a function is differentiable at any point, it is guaranteed to be continuous there, but the converse need not always hold true.

Mathematical reasoning allows us to make logical connections between statements and to understand the implications of given conditions. It involves the use of logic in mathematics to move from one set of facts to another.

In our exercise, the reasoning process involves examining the implications of differentiability and continuity for a real-valued function. By scrutinizing these properties and how they interrelate through the lens of conditional statements, we can derive meaningful conclusions about the function's behavior.

This capability is essential for developing proofs, for hypothesizing the behavior of functions under given conditions, and for solving problems effectively. Strong mathematical reasoning skills enable us to move beyond rote memorization of formulas and to engage deeply with the 'why' and 'how' of mathematics, which—as we see in this exercise—can lead one to uncover the nuances and intricacies of conditional statements and the properties of functions.