Question about continuity of a function...?
Q: Find points where function is discontinuous:
Function is f(x)= x divided by modulus of x
Plz tell me its solution with step by step...?
Plz tell me as soon as possible...?
The continuity of a function, at a certain point, happens when the left hand limit of the function is equal to the right hand limit of the function at that point. These limits must be equal to the value of function at that point.
Notice that the questionable point is x = 0.
`lim_(x-gt0)x/|x| = lim_(x-gt0) (xlt0)1/|-1| = -1` (use l'Hospital's theorem)
`lim_(x-gt0)x/|x| = lim_(x-gt0) (xgt0)1/|1| = 1`
Since the left hand limit `!=` right hand limit, the function is discontinuous at x = 0.
so just stick to the 3 conditions in the continuity test. a function is continuous at "x=c" if all of 3 are satisfied:
1) f(c) must exist
2) lim f(x) when x approaches c must exist
3) lim f(x) when x approaches c = f(c)
so, the function f(x)=x/|x| is discontinuous when 1 of those conditions is not satisfied.
- f(x)=x/|x| does not exist when the denominator (is |x|) equals 0.
=> when x = 0 , the function is discontinuous.
=> at every points else where, it's continuous (since the condition 2 and 3 are all satisfied with every points except for the point 0). so the interval on which the given function is continuous is
(-infinity, 0) U (0, infinity)
hope this helps and soon enough.