Continuity and Discontinuity of Functions

Domain of Functions Involving Tangent and Exponential Functions

When analyzing the domain and continuity of functions such as f(x) = e^{-tan^2(x)}, it is essential to consider where the function is defined and where it may be discontinuous.

• Domain: The function tan(x) is undefined where cos(x) = 0, i.e., at x = (2k+1)\frac{\pi}{2} for all integers k.

• Therefore, f(x) = e^{-tan^2(x)} is undefined at these points, and the domain is all real numbers except x = (2k+1)\frac{\pi}{2}.

Example: For x = \frac{\pi}{2}, tan(x) is undefined, so f(x) is also undefined.

Points of Discontinuity

Discontinuities occur exactly where tan(x) is undefined:

• At x = (2k+1)\frac{\pi}{2}, tan(x) is not defined, so f(x) is not defined.

Classification of Discontinuities

Discontinuities can be classified as removable or non-removable:

• Removable Discontinuity: If the limit exists as x approaches the point, but the function is not defined or does not match the limit.

• Non-removable Discontinuity: If the limit does not exist as x approaches the point.

For f(x) = e^{-tan^2(x)}, the discontinuities at x = (2k+1)\frac{\pi}{2} are non-removable because tan(x) approaches \pm\infty from either side, so the limit does not exist.

Intermediate Value Theorem (IVT)

Statement of the IVT

The Intermediate Value Theorem states that if f(x) is continuous on the interval [a, b] and N is any number between f(a) and f(b), then there exists at least one c in (a, b) such that f(c) = N.

Application of IVT

• To use the IVT, verify that the function is continuous on the interval.

• Check that the value N lies between f(a) and f(b).

• Conclude that there exists a c in (a, b) such that f(c) = N.

Example: Using IVT with a Complicated Function

Suppose you are given a function involving trigonometric and exponential terms, and you are asked to show that a certain value is achieved for some x using the IVT. The steps are:

1. Define the function and the interval.

2. Show the function is continuous on the interval (e.g., by showing it is composed of continuous functions and the interval avoids points of discontinuity).

3. Evaluate the function at the endpoints to check the sign change or that the target value is between them.

4. Apply the IVT to conclude the existence of a solution.

Key Formulae

• Tangent Function Undefined: is undefined at for all integers .

• Exponential of Tangent Squared:

• Intermediate Value Theorem: If is continuous on and is between and , then such that .

Table: Classification of Discontinuities for