- What is the difference between continuous and discontinuous piecewise functions?
- Is a straight line differentiable?
- Is a function differentiable at a removable discontinuity?
- Can a discontinuous function be integrated?
- How do you know if a function is discontinuous?
- Is a function differentiable at a corner?
- What function is continuous but not differentiable?
- Do discontinuous functions have Antiderivatives?
- How do you know if a graph is discontinuous?
- Where is a function discontinuous on a graph?
- Do all continuous functions have Antiderivatives?
- Is every continuous function is integrable?
- Can you differentiate a discontinuous function?
- Can a discontinuous function have a limit?
- Is a function discontinuous at a hole?
- How many types of improper integrals are there?
- Can functions be discontinuous?
- Is every continuous function differentiable?
- What are the 4 types of discontinuity?

## 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.