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It means that the function value doesn't make sudden jumps. For example, a function that rounds a number down to the closest integer is discontinuous for all integer values; for instance, when x changes from 0.99 to 1, or from 0.999999 to 1, or for any number arbitrarily close to 1 (but less than one) to one, the function value suddenly changes from 0 to 1. At other points, the function is continuous.

Q: What does it mean for a function to be continuous?

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yes it is a continuous function.

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.

It is a trigonometric function. It is also continuous.

Not at all.Y = x2 is a continuous function.

No, they can only be jump continuous.

Related questions

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

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

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.

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.

It is a trigonometric function. It is also continuous.

yes, every continuous function is integrable.

Not at all.Y = x2 is a continuous function.

No, they can only be jump continuous.

The tan [tangent] function.When a function has two or more brakes, this is not a continuous function, but it can be a continuous function in some intervals such as the tangent does.