Why aren't all functions continuous on their domain? Give example

Answer 1

A example of a function that is not continuous on its domain is given by a piecewise function.

For example f(x) = { x+4, when x <= 0, x+5 when x > 0}. The function has a value at x = 0, f(0) = 4, so 0 is in the domain of the function. But it is not continuous at 0 because the limit does not exist as x-> 0 -- different values from left and right.

Answer 2

If the function has the variable in the denominator, the values which make that denominator zero are NOT allowed in the domain of the function.

EX. f(x) = 1/(x-3) ----> 3 is not allowed in the domain because it makes the denominator zero

If the function has the variable under the square root, the values which make that square root negative are NOT allowed in the domain of the function

Ex. f(x) = sqrt( x+ 4) --> x+4>=0 ----> x>= -4

If the function of x is passed to the log function, the value which make that function zero or negative are

NOT allowed in the domain of the function

EX. f(x) = log(x+1) ---> x+1>0 ---> x > -1