Introduction to the Function

Consider the function f(x) 1/x. This function is a well-known example of a fundamental concept in calculus—discontinuity. Understanding the behavior of such a function is crucial for various applications in mathematics and engineering.

Defining the Function and Identifying Discontinuities

The function f(x) 1/x is not defined for x 0. At this point, the function approaches either positive or negative infinity, depending on the direction of approach. Therefore, the domain of f(x) is all real numbers except x 0.

Points of Discontinuity

At x 0:

For the function f(x) 1/x, the behavior as x approaches 0 from either side is critical.

As x approaches 0 from the positive side (x→0?), the function approaches positive infinity (f(x)→∞). As x approaches 0 from the negative side (x→0?), the function approaches negative infinity (f(x)→?∞).

Due to this behavior, there is a vertical asymptote at x 0, indicating an infinite discontinuity.

Further Analysis: Discontinuities within Integral Values

Discontinuity at x n where n is an integer:

Consider the function g(x) 1/[x], where [x] represents the greatest integer function, also known as the floor function. This function has a discontinuity at all integral values of x.

Example Analysis for g(x) 1/[x]

For example, when x approaches an integer from the left, the value of [x] is one less than the integer, causing the function to behave discontinuously. Similarly, when x approaches the integer from the right, the value of [x] is the integer, leading to a significant change in the value of the function.

Formally, the function g(x) is defined as:

g(x) 1/n for x in the interval [n, n 1), where n is an integer and n ≠ 0.

Hence, the function g(x) is discontinuous at all integral values of x except x 0.

Conclusion

In summary, the function f(x) 1/x is discontinuous at x 0, where it approaches infinity on either side, resulting in an infinite discontinuity. The function g(x) 1/[x] is discontinuous at all integer values, highlighting the importance of understanding the nature of the floor function and its implications on function behavior.

Key Takeaways:

The function f(x) 1/x is continuous on the intervals (?∞, 0) and (0, ∞). The discontinuity at x 0 is an infinite discontinuity. The function g(x) 1/[x] has discontinuities at all integer values except x 0.

Graphical Representation:

For a more intuitive understanding, graphing the function can help visualize these discontinuities. The graph of f(x) 1/x will show vertical asymptotes at x 0, while g(x) 1/[x] will exhibit discontinuities at each integer value of x.