91Ó°ÊÓ

Continuity is a fundamental concept in calculus and describes how a function behaves at a particular point without any jumps or breaks. For a function to be continuous at a point, say \( x = a \), three conditions must be met:

• The function \( f(x) \) exists at \( x = a \).
• The limit of \( f(x) \) as \( x \) approaches \( a \) exists.
• The limit of \( f(x) \) as \( x \) approaches \( a \) equals \( f(a) \).

If any of these conditions fail, the function is not continuous at that point. In the context of the exercise, the left-hand and right-hand limits are different as \( x \) approaches \( \frac{\pi}{2} \), meaning the function \( f \) is not continuous at this point. This is reflected in the conclusion of step 6 in the solution.

Differentiability is a condition that tells us whether a function has a derivative at a particular point. For a function to be differentiable at a point \( x = a \), it must be continuous at that point.
In other words, continuity is a prerequisite for differentiability. If a function is not continuous at a point, it cannot be differentiable there.
Mathematically, to check differentiability, we can use the definition:

• A function \( f(x) \) is differentiable at \( x = a \) if the limit \[ \lim_{{h \to 0}} \frac{f(a+h) - f(a)}{h} \] exists.

Since the function \( f \) in the exercise is not continuous at \( x = \frac{\pi}{2} \), it is also not differentiable, confirming the finding that \( f \) is neither continuous nor differentiable at that point.

Understanding limits is key to grasping both continuity and differentiability. A limit describes the behavior of a function as the input approaches a particular value. By calculating left-hand limits (approaching from the left) and right-hand limits (approaching from the right), we gain insights into how the function behaves.
In this particular exercise, we are interested in the limits as \( x \) approaches \( \frac{\pi}{2} \).

• The left-hand limit (LHL) is found by looking at values slightly less than \( \frac{\pi}{2} \), giving us a result of 3.
• The right-hand limit (RHL) considers values slightly greater, resulting in \( \frac{1}{3} \).

When these two limits are not equal, the overall limit does not exist, indicating discontinuity at that point.

Trigonometric functions, such as sine and cosine, are often involved in calculus problems due to their periodic nature and interesting properties. The function considered in the exercise involves \( \sin x \), which is a common trigonometric function defined for all real numbers.
Its range is \([-1, 1]\), and it defines periodic oscillating behavior over intervals. In the exercise, we compute expressions like \( \sin x - \sin^3 x \), where

• \( \sin^3 x = (\sin x)(\sin x)(\sin x) \).

These expressions lead us to insights about the function's behavior near specific points like \( \frac{\pi}{2} \). Understanding the distinctive properties of trigonometric functions, particularly around special angles like \( \frac{\pi}{2} \), can help in solving calculus problems effectively. In this case, since \( \sin(\frac{\pi}{2}) = 1 \), it was crucial for determining the behavior, limits, and continuity of the function in our analysis.