Existence of Laplace Transform of Piecewise Functions

[taxidriverhk]

Homework Statement

Let f(t) = t if 0<t<3
e^t if t>3

a. Is f(t) piece-wise continuous?
b. Is f(t) of exponential order α? Either prove it by producing an M, T and α that satisfies the definition, or show that no such constants exist.
c. Does the Laplace transform of f(t) exist? Briefly explain your answer.

Homework Equations

None

The Attempt at a Solution

I know it is not piecewise continuous already.
But can this point prove that the Laplace transform of this function does not exist?
Or do I still have to prove if it is of exponential order α? But I don't know how to find the M, α and T

Hope anyone can help me, thank you so much.

 

[LCKurtz]

Homework Statement

Let f(t) = t if 0<t<3
e^t if t>3

a. Is f(t) piece-wise continuous?
b. Is f(t) of exponential order α? Either prove it by producing an M, T and α that satisfies the definition, or show that no such constants exist.
c. Does the Laplace transform of f(t) exist? Briefly explain your answer.

I know it is not piecewise continuous already.

You do?

 

[taxidriverhk]

You do?

Sure, the limits of e^t and t as t approaches 3 are not equal, so f(t) should not be piece-wise continuous, isn't that right?

 

[LCKurtz]

Sure, the limits of e^t and t as t approaches 3 are not equal, so f(t) should not be piece-wise continuous, isn't that right?

No, that isn't right. That would make the functions continuous. What is the definition of piecewise continuous given in your text?