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.

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.

A continuous linear decreasing function is a line that goes on forever and has a negative slope (is downhill from left to right).

For example, the line y = -x is a continuous linear decreasing function.

Yes.

Yes, it does "appear" to be continuous, by the simple fact that it is continuous for all values of the input.

no , since there is a function which lsc but not usc.

The graph of a continuous function will not have any 'breaks' or 'gaps' in it.

You can draw it without lifting your pencil or pen.

The graph of a discrete function will just be a set of lines.

All differentiable functions need be continuous at least.

fist disply your anser

No. y = 1/x is continuous but unbounded.

Yes.

Yes, a corner is continuous, as long as you don't have to lift your pencil up then it is a continuous function. Continuous functions just have no breaks, gaps, or holes.

It is a continuous function.

If the line is a straight line, it is a linear function.

Both are polynomials. They are continuous and are differentiable.

continuous

The numerator function x2 - 4 and the denominator function x2 + 3x + 2 are both continuous functions of x for the entire x-axis. However, the quotient of these two functions is not continuous when the denominator function has the value of 0, because division by zero is not defined. The denominator function is 0 when x = -1 or -2. Therefore, the quotient function is not fully continuous over any intervals that include -1 or -2, but it is "piecewise continuous" over other intervals of the x-axis.

There exists an N such that for all n>N, for any x. Now let n>N, and consider the continuous function . Since it is continuous, there exists a such that if , then . Then so the function f(x) is continuous.

Υou show that it is continuous in every element of it's domain.

It is not necessarily so.

No.

Demand schedule: a list of demand/price equivalencies. It can best be seen as a table with discrete points.

Demand function: a continuous function of price-demand interaction.

Main difference: schedule is discrete; function is continuous.

A function that is continuous over a finite closed interval must have both a maximum and a minimum value on that interval, according to the Extreme Value Theorem. This theorem states that if a function is continuous on a closed interval ([a, b]), then it attains its maximum and minimum values at least once within that interval. Therefore, it is impossible for a continuous function on a finite closed interval to not have a maximum or minimum value.

continuous

The function is not continuous.

No. Not all functions are continuous. For example, the function f(x) = 1/x is undefined at x = 0.

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.

How about f(x) = floor(x)? (On, say, [0,1].) It's monotone and therefore of bounded variation, but is not Lipschitz continuous (or even continuous).

Four discrete points do not define a continuous function.

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.

A continuous linear function produces a straight line graph that can be extended indefinitely in either direction.

If the two ordered pairs are plotted on a graph then a straight line can be drawn joining these points. If that line is extended beyond both ends then there are no set limits and the function becomes continuous.

It is a function which is usually used with continuous distributions, to give the probability associated with different values of the variable.

the maximum or minimum value of a continuous function on a set.

No. It has a discontinuity at every integer value.

Well obviously it's for knowing what's happening in the past.

A cadlag is a mathematical term for a function which is right continuous and has a left limit.

produce smooth, continuous muscle contraction

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.

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".

Apparently f(x)=sqrt(x)

But I'm not sure why. That's what I'm looking for now :)

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.