Question: Can A Discontinuous Function Be Differentiable?

What is the difference between continuous and discontinuous piecewise functions?

The piecewise function shown in this example is continuous (there are no “gaps” or “breaks” in the plotting).

Piecewise defined functions may be continuous (as seen in the example above), or they may be discontinuous (having breaks, jumps, or holes as seen in the examples below)..

Is a straight line differentiable?

If a function f is differentiable at its entire domain, that simply means that you can zoom into each point, and it will resemble a straight line at each one (though, obviously, it can resemble a different line at each point – the derivative need not be constant). … (For all other x, of course, it is differentiable).

Is a function differentiable at a removable discontinuity?

No. A function with a removable discontinuity at the point is not differentiable at since it’s not continuous at . Continuity is a necessary condition.

Can a discontinuous function be integrated?

We call functions whose discontinuities have Lebesgue measure zero piecewise continuous. Discontinuous functions can be integrable, although not all are. … This function cannot be integrated. However, if we consider between and , it seems clear that the entire area in the unit square does lie “under” the curve.

How do you know if a function is discontinuous?

Start by factoring the numerator and denominator of the function. A point of discontinuity occurs when a number is both a zero of the numerator and denominator. Since is a zero for both the numerator and denominator, there is a point of discontinuity there. To find the value, plug in into the final simplified equation.

Is a function differentiable at a corner?

A function is not differentiable at a if its graph has a corner or kink at a. … Since the function does not approach the same tangent line at the corner from the left- and right-hand sides, the function is not differentiable at that point.

What function is continuous but not differentiable?

In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is an example of a fractal curve. It is named after its discoverer Karl Weierstrass.

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 a graph is discontinuous?

If the function factors and the bottom term cancels, the discontinuity at the x-value for which the denominator was zero is removable, so the graph has a hole in it. After canceling, it leaves you with x – 7. Therefore x + 3 = 0 (or x = –3) is a removable discontinuity — the graph has a hole, like you see in Figure a.

Where is a function discontinuous on a graph?

We say the function is discontinuous when x = 0 and x = 1. There are 3 asymptotes (lines the curve gets closer to, but doesn’t touch) for this function. They are the x-axis, the y-axis and the vertical line x=1 (denoted by a dashed line in the graph above).

Do all continuous functions have Antiderivatives?

Every continuous function has an antiderivative, and in fact has infinitely many antiderivatives. Two antiderivatives for the same function f(x) differ by a constant.

Is every continuous function is integrable?

If f is continuous everywhere in the interval including its endpoints which are finite, then f will be integrable. … A function is continuous at x if its values sufficiently near x are as close as you choose to one another and to its value at x .

Can you differentiate a discontinuous function?

So differentiable implies continuous. In other words, discontinuous implies not differentiable. Technical point: you do require one fact about limits here: that the product of two limits (both of which exists) is the limit of the product (which is guaranteed to exist).

Can a discontinuous function have a limit?

All discontinuity points are divided into discontinuities of the first and second kind. There exist left-hand limit limx→a−0f(x) and right-hand limit limx→a+0f(x); These one-sided limits are finite.

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.

How many types of improper integrals are there?

two typesThere are two types of improper integrals: The limit a or b (or both the limits) are infinite; The function f(x) has one or more points of discontinuity in the interval [a,b].

Can functions be discontinuous?

Discontinuous functions are functions that are not a continuous curve – there is a hole or jump in the graph. It is an area where the graph cannot continue without being transported somewhere else.

Is every continuous function 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.

What are the 4 types of discontinuity?

There are four types of discontinuities you have to know: jump, point, essential, and removable.