1. What Are The Trigonometric Functions That Are Always Continuous? 2. For The Function F(x) = [x], Where [x] Represents The Greatest Integer Function, Determine The Points At Which It Is Continuous. Options Are: A) X = 1 B) X = 1.5 C) X = -2 3. Describe The Continuity Of The Function F(x) = |x| + |x - 1| At X = 0.

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In the realm of mathematics, particularly in trigonometry and calculus, the concept of continuity is fundamental. A function is said to be continuous if its graph can be drawn without lifting the pen from the paper, meaning there are no breaks, jumps, or holes. When discussing trigonometric functions, which are the bedrock of understanding periodic phenomena, identifying which functions exhibit this continuous behavior is crucial.

Trigonometric functions, such as sine, cosine, tangent, cotangent, secant, and cosecant, describe the relationships between the angles and sides of triangles. Among these, two stand out for their unwavering continuity across their entire domain: the sine function and the cosine function. Let's delve into why these functions are always continuous and why this property is so significant.

Sine and Cosine: The Epitome of Continuity

The sine function, denoted as sin(x), and the cosine function, denoted as cos(x), are defined based on the unit circle. For any angle x, sin(x) represents the y-coordinate of the point on the unit circle corresponding to that angle, while cos(x) represents the x-coordinate. This geometric definition inherently lends itself to continuity. As the angle x varies smoothly, the x and y coordinates also change smoothly, resulting in continuous graphs.

The graphs of sin(x) and cos(x) are smooth, undulating curves that oscillate between -1 and 1. There are no abrupt changes, no vertical asymptotes, and no gaps. This visual representation underscores their continuous nature. Mathematically, we can express this continuity using limits. For any real number a:

  • lim (x→a) sin(x) = sin(a)
  • lim (x→a) cos(x) = cos(a)

These equations state that the limit of the function as x approaches a is equal to the function's value at a, which is the very definition of continuity at a point. Since this holds true for all real numbers a, sine and cosine are continuous everywhere.

Why Continuity Matters

The continuity of sine and cosine has far-reaching implications in various fields:

  • Physics: These functions are used to model periodic phenomena like oscillations, waves, and simple harmonic motion. The continuous nature of sine and cosine ensures that these models accurately represent the smooth, flowing nature of these phenomena.
  • Engineering: In signal processing, electrical circuits, and mechanical systems, sine and cosine are used to analyze and design systems that behave predictably and without sudden disruptions. Continuity is vital for ensuring the stability and reliability of these systems.
  • Calculus: Continuity is a prerequisite for many important theorems in calculus, such as the Intermediate Value Theorem and the Extreme Value Theorem. These theorems allow us to make powerful deductions about the behavior of continuous functions.

The Discontinuous Trigonometric Functions

While sine and cosine enjoy universal continuity, other trigonometric functions are not so fortunate. Tangent (tan(x)), cotangent (cot(x)), secant (sec(x)), and cosecant (csc(x)) all have points of discontinuity. These discontinuities arise because these functions are defined as ratios involving sine and cosine, and division by zero leads to undefined points.

For example, tan(x) = sin(x) / cos(x), so it is discontinuous wherever cos(x) = 0, which occurs at x = (π/2) + nπ, where n is an integer. Similarly, cot(x) = cos(x) / sin(x) is discontinuous where sin(x) = 0. Secant and cosecant, being the reciprocals of cosine and sine respectively, also inherit discontinuities from the zeros of sine and cosine.

Conclusion

The sine and cosine functions stand as pillars of continuity within the world of trigonometry. Their smooth, unbroken graphs and their adherence to the limit definition of continuity make them indispensable tools in mathematics, physics, engineering, and beyond. Understanding their continuous nature is not just an academic exercise; it's a gateway to comprehending and modeling the continuous, flowing phenomena that permeate our world. While other trigonometric functions have their own unique properties and applications, it is the unwavering continuity of sine and cosine that earns them a special place in the mathematical landscape.

In the captivating world of mathematical functions, continuity serves as a fundamental concept that distinguishes functions with smooth, unbroken graphs from those with abrupt jumps or breaks. The function f(x) = [x], where [x] represents the greatest integer less than or equal to x, provides an intriguing example to explore the nuances of continuity. This function, often referred to as the floor function, maps a real number to the largest integer that is less than or equal to that number. Our exploration will determine the points at which this function exhibits continuity.

To dissect the continuity of the floor function, we must first understand its behavior. For any integer n, the floor function returns n itself: [n] = n. However, for any non-integer value x between two consecutive integers n and n+1, the floor function returns n. This creates a step-like pattern in the graph of the function, where the value remains constant within each interval [n, n+1) and then jumps to the next integer at x = n+1.

Analyzing Continuity at Integer Points

Let's consider an integer value, say x = 1. To assess continuity at this point, we need to examine the left-hand limit, the right-hand limit, and the function's value at x = 1:

  • Left-hand limit: lim (x→1-) [x] = 0. As x approaches 1 from the left, it takes values slightly less than 1, so [x] is 0.
  • Right-hand limit: lim (x→1+) [x] = 1. As x approaches 1 from the right, it takes values slightly greater than 1, so [x] is 1.
  • Function value: f(1) = [1] = 1.

For a function to be continuous at a point, the left-hand limit, the right-hand limit, and the function value must all be equal. In this case, the left-hand limit (0) is not equal to the right-hand limit (1) or the function value (1). Therefore, the floor function is discontinuous at x = 1.

This same reasoning applies to any integer value. At each integer n, the floor function jumps from n-1 to n, creating a discontinuity. The left-hand limit will be n-1, while the right-hand limit and the function value will both be n.

Examining Continuity at Non-Integer Points

Now, let's consider a non-integer value, such as x = 1.5. Again, we examine the left-hand limit, the right-hand limit, and the function value:

  • Left-hand limit: lim (x→1.5-) [x] = 1. As x approaches 1.5 from the left, it takes values slightly less than 1.5, so [x] is 1.
  • Right-hand limit: lim (x→1.5+) [x] = 1. As x approaches 1.5 from the right, it takes values slightly greater than 1.5 but still less than 2, so [x] is 1.
  • Function value: f(1.5) = [1.5] = 1.

In this instance, the left-hand limit, the right-hand limit, and the function value are all equal to 1. This indicates that the floor function is continuous at x = 1.5.

This behavior holds true for any non-integer value. For any x that is not an integer, there exists an open interval around x within which the floor function remains constant. Therefore, the left-hand limit, the right-hand limit, and the function value will all be the same, ensuring continuity.

Generalizing the Continuity of the Floor Function

Based on our analysis, we can conclude that the floor function, f(x) = [x], is continuous at all non-integer points and discontinuous at all integer points. This is because the function