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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.
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.
Is a cosine function continuous?
Yes. The cosine function is continuous. The sine function is also continuous. The tangent function, however, is not 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.
Is y equals 0 a continuous function?
By the definition of continuity, since the limit and f(x) both exist and are equal (to 0) at each value of x, y=0 is continuous. This is true for any constant function.
Does the cosine function appear to be continuous?
yes it is a continuous function.
A polynomial function is always continuous?
Yes, a polynomial function is always continuous
Every continuous function is integrible but converse is not true every integrable function is not continuous?
That's true. If a function is continuous, it's (Riemman) integrable, but the converse is not true.
What function is continuous everywhere but not differentiable?
Weistrass function is continuous everywhere but not differentiable everywhere
Can a function have a limit at every x-value in its domain?
Yes, that happens with any continuous function. The limit is equal to the function value in this case.Yes, that happens with any continuous function. The limit is equal to the function value in this case.Yes, that happens with any continuous function. The limit is equal to the function value in this case.Yes, that happens with any continuous function. The limit is equal to the function value in this case.
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.
Let f be an odd function with antiderivative F. Prove that F is an even function. Note we do not assume that f is continuous or even integrable.?
An antiderivative, F, is normally defined as the indefinite integral of a function f. F is differentiable and its derivative is f.If you do not assume that f is continuous or even integrable, then your definition of antiderivative is required.
Is continuous signals has no amplitudes?
By definition a continuous signal is just that continuous to have no amplitude is to mean it doesn't exists
