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A piecewise continuous function is a function that is composed of multiple continuous pieces, with each piece defined on a specific interval. The function may have discontinuities at the endpoints of each interval, but within each interval, the function is continuous.
Exponential order is a property of a function that describes how quickly the function grows or decays as its input increases. A function f(t) is said to be of exponential order if there exist constants M and c such that |f(t)| ≤ Me^ct for all values of t. This means that the function grows or decays at most exponentially as its input increases.
Knowing that a function is of exponential order allows us to make useful mathematical statements and predictions about its behavior. For example, we can use this property to determine the rate at which the function grows or decays, or to evaluate certain integrals involving the function.
Some common examples of functions that are of exponential order include exponential functions, trigonometric functions, and polynomial functions. For instance, f(t) = e^t, f(t) = sin(t), and f(t) = t^2 are all of exponential order.
Yes, it is possible for a function to be both piecewise continuous and of exponential order. For example, the function f(t) = e^t is both piecewise continuous and of exponential order, as it is continuous on the entire real number line and satisfies the definition of exponential order.