To determine whether a
piecewise function is continuous or discontinuous, in addition to checking the boundary points, we must also check whether each of the functions that make up the piecewise function is continuous.
Given a piecewise function, determine whether it is continuous.
Determine whether each component function of the piecewise function is continuous. If there are discontinuities, do they occur within the domain where that component function is applied?
For each boundary point
of the piecewise function, determine if each of the three conditions hold.
Determining whether a piecewise function is continuous
Determine whether the function below is continuous. If it is not, state the location and type of each discontinuity.
The two functions composing this piecewise function are
on
and
on
The sine function and all polynomial functions are continuous everywhere. Any discontinuities would be at the boundary point,
At
let us check the three conditions of continuity.
Condition 1:
Because all three conditions are not satisfied at
the function
is discontinuous at
A continuous function can be represented by a graph without holes or breaks.
A function whose graph has holes is a discontinuous function.
A function is continuous at a particular number if three conditions are met:
Condition 1:
exists.
Condition 2:
exists at
Condition 3:
A function has a jump discontinuity if the left- and right-hand limits are different, causing the graph to “jump.”
A function has a removable discontinuity if it can be redefined at its discontinuous point to make it continuous. See
[link] .
Some functions, such as polynomial functions, are continuous everywhere. Other functions, such as logarithmic functions, are continuous on their domain. See
[link] and
[link] .
For a piecewise function to be continuous each piece must be continuous on its part of the domain and the function as a whole must be continuous at the boundaries. See
[link] and
[link] .
Section exercises
Verbal
State in your own words what it means for a function
to be continuous at
Informally, if a function is continuous at
then there is no break in the graph of the function at
and
is defined.
the transfer of energy by a force that causes an object to be displaced; the product of the component of the force in the direction of the displacement and the magnitude of the displacement
A wave is described by the function D(x,t)=(1.6cm) sin[(1.2cm^-1(x+6.8cm/st] what are:a.Amplitude b. wavelength c. wave number d. frequency e. period f. velocity of speed.
A body is projected upward at an angle 45° 18minutes with the horizontal with an initial speed of 40km per second. In hoe many seconds will the body reach the ground then how far from the point of projection will it strike. At what angle will the horizontal will strike
Suppose hydrogen and oxygen are diffusing through air. A small amount of each is released simultaneously. How much time passes before the hydrogen is 1.00 s ahead of the oxygen? Such differences in arrival times are used as an analytical tool in gas chromatography.
the science concerned with describing the interactions of energy, matter, space, and time; it is especially interested in what fundamental mechanisms underlie every phenomenon
