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There are several different ways of defining continuity. The following is based on work done by Bolzano and Weierstrass.

A function f(x), of a variable x is continuous at the point c if, given any positive number e, however small, it is possible to find d such that

f(c) - e < f(c) < f(c) + e

for ALL x in c - d < x < c + d.

In simpler terms, it is possible to find an interval around x such that for ALL values of x' in that interval, the value of the function, f(x'), is close to f(x).

Determining continuity visually, it is easy: if the function can be drawn without lifting your pencil, then it is continuous and if you cannot, it is not.

Q: What is the definition of continuity of the function and how do you determine a function is continuous?

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A function f is continuous at c if:f(c) is defined.lim "as x approaches c" f(x) exists.lim "as x approaches c" f(x) = f(c).

If you are looking at a graph and you want to know if a function is continuous, ask yourself this simple question: Can I trace the graph without lifting my pencil? If the answer is yes, then the function is continuous. That is, there should be no "jumps", "holes", or "asymptotes".

An intuitive answer (NOTE: this is far from precise!) A function is continuous if you can trace its graph without lifting your pencil from the page. If, additionally, it is smooth everywhere without any jagged edges or abrupt corners, then it is differentiable. It is not possible for a function to be differentiable but not continuous. On the other hand, plenty of functions are continuous without being differentiable.

Sure! The definition of Laplace transform involves the integral of a function, which always makes discontinuous continuous.

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.

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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.

A function f is continuous at c if:f(c) is defined.lim "as x approaches c" f(x) exists.lim "as x approaches c" f(x) = f(c).

If you are looking at a graph and you want to know if a function is continuous, ask yourself this simple question: Can I trace the graph without lifting my pencil? If the answer is yes, then the function is continuous. That is, there should be no "jumps", "holes", or "asymptotes".

The way I understand it, a continuos function is said not to be "uniformly continuous" if for a given small difference in "x", the corresponding difference in the function value can be arbitrarily large. For more information, check the article "Uniform continuity" in the Wikipedia, especially the examples.

An intuitive answer (NOTE: this is far from precise!) A function is continuous if you can trace its graph without lifting your pencil from the page. If, additionally, it is smooth everywhere without any jagged edges or abrupt corners, then it is differentiable. It is not possible for a function to be differentiable but not continuous. On the other hand, plenty of functions are continuous without being differentiable.

Sure! The definition of Laplace transform involves the integral of a function, which always makes discontinuous continuous.

Yes. The cosine function is continuous. The sine function is also continuous. The tangent function, however, is not continuous.

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.

yes it is a continuous function.

Yes, a polynomial function is always continuous

That's true. If a function is continuous, it's (Riemman) integrable, but the converse is not true.

Weistrass function is continuous everywhere but not differentiable everywhere