Question

Determine the largest interval over which the given function is continuous. $$ f(x)=\operatorname{sech} x $$

Short answer

The function \( \operatorname{sech}(x) \) is continuous over \( (-\infty, \infty) \).

Step-by-step solution

Understand the Hyperbolic Secant

The hyperbolic secant function, denoted as \( \operatorname{sech}(x) \), is defined as \( \operatorname{sech}(x) = \frac{1}{\cosh(x)} \) where \( \cosh(x) = \frac{e^x + e^{-x}}{2} \). It is continuous wherever \( \cosh(x) \) is not zero, because a division by zero would make the function undefined.

Examine Cosh(x) for Zeros

The function \( \cosh(x) \) is given by \( \frac{e^x + e^{-x}}{2} \). This function is always positive because both \( e^x \) and \( e^{-x} \) are positive for all real \( x \). Thus, \( \cosh(x) \) has no zeros for any real number \( x \).

Determine Continuity of Sech(x)

Since \( \cosh(x) \) is positive everywhere on the real line, the function \( \operatorname{sech}(x) = \frac{1}{\cosh(x)} \) is defined, continuous, and differentiable for all real \( x \).

Conclude the Interval of Continuity

Since \( \operatorname{sech}(x) \) is defined and continuous wherever \( \cosh(x) \) does not equal zero and since \( \cosh(x) eq 0 \) for any real \( x \), the interval of continuity for \( \operatorname{sech}(x) \) is \( (-\infty, \infty) \).

Key concepts

Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions but for the hyperbola instead of the circle. They include functions like hyperbolic sine (\( \sinh(x) \)), hyperbolic cosine (\( \cosh(x) \)), and hyperbolic tangent (\( \tanh(x) \)), among others. These functions are denoted with an 'h', differentiating them from their trigonometric counterparts.
To understand hyperbolic functions, it helps to visualize the shape of a hyperbola just as you would for a circle with trigonometric functions. They have important properties and applications, especially in fields like calculus and engineering, as they relate closely to exponential functions.
For instance, the hyperbolic secant (\( \operatorname{sech}(x) \)) is defined as the reciprocal of the hyperbolic cosine, which can be expressed as \( \operatorname{sech}(x) = \frac{1}{\cosh(x)} \). Each hyperbolic function is continuous and smooth, making them an essential tool in analyzing certain mathematical problems.

Continuous Functions

In mathematics, a continuous function is one where small changes in the input result in small changes in the output. Imagine drawing a function's graph without ever lifting your pencil – that's continuity.
A function is continuous at a point \( x = a \) if the limit of the function as \( x \) approaches \( a \) is equal to the function's value at \( a \). In simpler terms, there are no jumps, breaks, or holes.
The hyperbolic secant function \( \operatorname{sech}(x) \) is continuous across its domain, provided \( \cosh(x) \) never equals zero. Since \( \cosh(x) \) is always positive for real numbers \( x \), \( \operatorname{sech}(x) \) is continuous everywhere on the real number line. This means that \( \operatorname{sech}(x) \) smoothly transits from one point to the next without any interruptions.

Real Number Line

The real number line is a complete and continuous set of numbers that extends infinitely in both positive and negative directions.
On this line, every point corresponds to a unique real number, stretching from negative infinity (\(-\infty\)) to positive infinity (\(\infty\)). Every conceivable real number is on this line, whether they are integers, fractions, or irrational numbers like the square root of 2.
When we discuss the domain of a function like \( \operatorname{sech}(x) \), which is defined in terms of real numbers, we are often referencing how the function behaves across the entire real number line. Since \( \cosh(x) \) has no zeros across the real numbers, \( \operatorname{sech}(x) \) remains continuous for all \( x \) from \(-\infty\) to \(\infty\). Thus, its domain includes the entire real number line, something that makes it especially robust for various applications in math and science.