Sage-Tips

Can a composition of discontinuous functions be continuous?

Composition of a continuous function and a discontinuous function, can be continous.

Can a composite function be continuous?

In particular rational functions are continuous at all points where the denominator is zero. Theorem (Composite functions) Assume that f is continuous at a and g is continuous at b = f(a). then the composite function h = g ◦ f is continuous at a. Hence h = g ◦ f is continuous at a.

Is the composite of two continuous functions continuous?

The composition of continuous functions is also continuous. So, if f(x) and g(x) are continuous functions, meaning that they are continuous at all points at which they are defined, then f(g(x)) is also continuous. Notice that, since f(x) and g(x) in this example are both polynomials, they are both continuous.

Can a function be continuous and not defined?

2) a function can only be continuous on its domain, since the definition involves evaluating f at a (in particular if f(a) is not defined, then f(x) cant be continuous at a); 3) a continuous function has a limit at a (in particular, if limx→a f(x) does not exist, f cant be continuous).

Can a discontinuous function be defined?

A discontinuous function is the opposite. It is a function that is not a continuous curve, meaning that it has points that are isolated from each other on a graph. When you put your pencil down to draw a discontinuous function, you must lift your pencil up at least one point before it is complete.

How do you prove f is continuous on an interval?

A function is said to be continuous on an interval when the function is defined at every point on that interval and undergoes no interruptions, jumps, or breaks. If some function f(x) satisfies these criteria from x=a to x=b, for example, we say that f(x) is continuous on the interval [a, b].

What type of functions are not continuous?

In other words, a function is continuous if its graph has no holes or breaks in it. For many functions it’s easy to determine where it won’t be continuous. Functions won’t be continuous where we have things like division by zero or logarithms of zero.

Is the sum of f+g and f*g always discontinuous?

Then both f and g are discontinuous everywhere, but f+g is 1 everywhere, and f*g is 0 everywhere. These two are clearly continuous. , Degree in CS. 20 years of self-taught knowledge in Math and complexity theory. If f (x) and g (x) are discontinuous, then the sum f (x) + g (x) may be either continuous or discontinuous.

How do you prove that a function is continuous or discontinuous?

Also if f (x) and and g (x) are discontinuous, the product f (x)*g (x) may be continuous or discontinuous. To prove discontinuous case, use the same definition for f (x) and g (x) given above. Then their product is 0 whenever x < 0 and is equal to r whenever x >= 0. So, the result is discontinuous.

What are the rules and formulas of continuity and discontinuity?

According to the Rules and formulas of Continuity and Discontinuity. Sum of two discontinuous function is always Discontinuous. 2. Product of two Discontinuous function is always is continuous. This same rule is applied for the Continuous function also. Should I hire remote software developers from Turing.com?

How do you know if a composition is continuous or continuous?

Then as long as the range of g (x) is inside that region where f (x) is continuous, then the composition f (g (x)) will be continuous. Meanwhile, if f (x) is discontinuous, then g (f (x)) will only be continuous if g (x) happens to map the values on both sides of the discontinuity back to the same value.