91Ó°ÊÓ

Continuity is a fundamental concept in calculus, describing a function that is unbroken or smooth across its domain. When we say a function is continuous at a point, we mean that there are no abrupt changes in its value around that point.
The mathematical definition of continuity at a point \(x = c\) states that for a function \(f(x)\):

• The function must be defined at \(c\), meaning \(f(c)\) exists.
• The limit of \(f(x)\) as \(x\) approaches \(c\) must exist.
• The value of the function at \(c\) equals the limit as \(x\) approaches \(c\), i.e., \(\lim_{{x \to c}} f(x) = f(c)\).

To put it simply, you can draw the graph of a continuous function without lifting your pencil.
In our exercise, proving that differentiability implies continuity utilizes these continuity principles. Differentiability ensures the limit exists consistently as \(h\) approaches zero, promoting a seamless transition of value at every point in the interval \((a, b)\).
Therefore, a differentiable function, like \(f(x)\), is naturally continuous at each point within its domain.

The derivative of a function is one of the central ideas in calculus. It provides a measurement of how a function's output changes as its input changes. In more intuitive terms, it represents the 'slope' or 'rate of change' of the function at a given point.
In mathematics, if you see \(f'(x)\), this denotes the derivative of the function \(f(x)\). It is formally defined as: \[f^{\prime}(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.\] This limit tells us the instantaneous rate at which \(f(x)\) changes with respect to \(x\).
In the given exercise, the presence of a derivative for each \(x\) in the interval \((a, b)\) implies the function is smooth and the rate of change is consistent. This is because the derivative exists, meaning the limit is real, finite, and consistent across the interval. It confirms no jumps or discontinuities occur, leading us to conclude the function is differentiable.
Differentiability encompasses more than just having a derivative. It implies additional smoothness, ensuring continuity across the interval \((a, b)\).

The concept of a limit is crucial in calculus and lays the groundwork for understanding both continuity and differentiability. A limit describes the behavior of a function as it approaches a particular point, capturing its trend or direction.
Mathematically, the limit of a function \(f(x)\) as \(x\) approaches a point \(a\) is denoted by: \[\lim_{x \to a} f(x) = L.\]This expression means that as \(x\) gets closer to \(a\), \(f(x)\) gets closer to \(L\).
Limits bridge the gap between the behavior of a function near a point and the value it takes at that point. For a function to be continuous, the limit as \(x\) approaches any given point must equal the function's value at that point.
In the scenario of our exercise, the limit was used to explain how differentiability leads to continuity. As \(h\) becomes infinitesimally small, \(f(x+h)\) converges towards \(f(x)\), meaning the function has no sudden leaps and aligns itself smoothly through every point in the interval \((a, b)\).