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Intuitively, a continuous function y = f(x) is one where small changes in x result in small changes in y.
More rigorously, consider the function y = f(x) defined on the domain D to the codomain C where both D and C are subsets of R.
Then f(x) is continuous at a point p in D if the limit of f(x) as x approaches p within D is f(p).
The function is said to be continuous is it is continuous at every point in its domain.
The domain and codomain of f can be extended to multiple dimensions provided a suitable metric (eg Pythagorean distance) is used.
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Can a discontinuous function have a laplace transform?
Sure! The definition of Laplace transform involves the integral of a function, which always makes discontinuous continuous.
What is the meaning of continuous first derivatives of a function?
A continuous function is one where there are no discontinuities or step changes in the function, i.e. for a small change in input value, as that small change approaches zero, there is a progressively smaller change in output value. There are many definitions, some formal and some intuitive, for continuous functions. The definition given above is intuitive. The same definition can be give to the deriviatives or the integrals of a function. Continousness does not depend on being a deriviative or integral.
Does the cosine function appear to be continuous?
yes it is a continuous function.
Is the infinite sum of continuous function continuous?
An infinite sum of continuous functions does not have to be continuous. For example, you should be able to construct a Fourier series that converges to a discontinuous function.
What is the sin function?
It is a trigonometric function. It is also continuous.
