Can A Discontinuous Function Have A Limit?

At what points is the function continuous?

Saying a function f is continuous when x=c is the same as saying that the function’s two-side limit at x=c exists and is equal to f(c)..

Do discontinuous functions have Antiderivatives?

Most functions you normally encounter are either continuous, or else continuous everywhere except at a finite collection of points. For any such function, an antiderivative always exists except possibly at the points of discontinuity.

How do you know if an integral exists?

In order to show that the integral exists, we check if the integrand function is continuous, positive and decreasing in the given integral limits.

How do you know if an integral converges?

If the integration of the improper integral exists, then we say that it converges. But if the limit of integration fails to exist, then the improper integral is said to diverge.

What is the difference between continuous and differentiable?

A continuous function is a function whose graph is a single unbroken curve. A discontinuous function then is a function that isn’t continuous. A function is differentiable if it has a derivative. You can think of a derivative of a function as its slope.

Can a graph be continuous but not differentiable?

In particular, any differentiable function must be continuous at every point in its domain. The converse does not hold: a continuous function need not be differentiable. For example, a function with a bend, cusp, or vertical tangent may be continuous, but fails to be differentiable at the location of the anomaly.

How do you know if a function is continuous or discontinuous?

We said above that if any of the three conditions of continuity is violated, function is said to be discontinuous. =>f(x) is discontinuous at –1. However, if we try to find the Limit of f(x), we conclude that f(x) is continuous on all the values other than –1.

Is a function discontinuous at a hole?

The function is not continuous at this point. This kind of discontinuity is called a removable discontinuity. Removable discontinuities are those where there is a hole in the graph as there is in this case. … In other words, a function is continuous if its graph has no holes or breaks in it.

Does a hole make a function discontinuous?

Discontinuous functions are functions that are not a continuous curve – there is a hole or jump in the graph.

Can a limit be discontinuous?

A finite discontinuity exists when the two-sided limit does not exist, but the two one-sided limits are both finite, yet not equal to each other.

Can a differentiable function be discontinuous?

It is possible for a differentiable function to have discontinuous partial derivatives. An example of such a strange function is f(x,y)={(x2+y2)sin(1√x2+y2) if (x,y)≠(0,0)0 if (x,y)=(0,0).

Can you integrate a discontinuous function?

Is every discontinuous function integrable? No. … It’s not integrable! For any partition of [0,1], every subinterval will have parts of the function at height 0 and at height 1, so there’ no way to make the Riemann sums converge.

How do you know if a limit is continuous or discontinuous?

A function being continuous at a point means that the two-sided limit at that point exists and is equal to the function’s value. Point/removable discontinuity is when the two-sided limit exists, but isn’t equal to the function’s value.