For what value of the constant c is the function f continuous on (-inf, inf)

Answer 1

this problem suggests a piece wise function such as for example

f(x) = 2x + c for x< 4

f(x) = 7x -c for x> 4

set the two expressions equal and solve for c

2x +c = 7x -c

2c = 5x

c =5x/2

plug in 4

c= 5(4)/2 = 10

c = 10 makes the function continuous

the two lines then meet at (4,18)

it helps to graph them

or another type of problem that arises is a line with a "hole" in it, a one point discontinuity

a function where the denominator = 0 at that point

f(x) = (x+2)(x-1)(x-1) is the same as the straight line y=x+2 but with a point missing at x=1, (1,3)

same situation with

f(x) = (x+2)(x-c)/(x-c) is a straight line with a "hole" in it at the point (c,2+c)

to remove the discontinuity make it a step wise function

f(x) = (x+2)(x-c)/(x-c) when x does not equal c

f(x) = x+2 when x=c

then the function is continuous for all real values of x