f(x) = { x ; x<= 0
x^2 ; 0 < x < =1
2/x ; 1 < x <= 2
x-1 ; x>2
My problem is whether f(x) is continuous at x = 0 or not. Because left hand limit = f(0) when x approaches to 0 but the right hand limit f(0) when x approaches to 0. is it continuous ?
f(x) = { x ; x<= 0
x^2 ; 0 < x < =1
2/x ; 1 < x <= 2
x-1 ; x>2
My problem is whether f(x) is continuous at x = 0 or not. Because left hand limit = f(0) when x approaches to 0 but the right hand limit f(0) when x approaches to 0. is it continuous ?
The left hand limit is 0, since the function is f(x) = x to the left of x = 0.
The right hand limit is 0, since the function is f(x) = x^2 to the right of x = 0.
Since the left hand limit and right hand limit are equal, the function is continuous at x = 0.
Sort of. Like I said, when dealing with limits, you do not worry about what the function equals at a point. All you want to know is what the function tends to NEAR that point.
I think you can see that as you get closer and closer to x = 0 on each side, you get closer and closer to f(x) = 0... Since this happens on both sides, the limit exists, and since the function is defined at that point and is equal to the limit, the function is continuous. Does that make sense?