f(x) = x + | x | is continuous for
(A) x∈(−∞,∞) (B) x∈(−∞,∞) −{0} (C) only x > 0 (D) no value of x
1since for x < 0 f(x) = 0 which is x - axis
and for x > 0 f(x) = 2x so
f(x) is continuous for all x
hence (a)
39Arey manmay...sum of two continuous functions(which are continuous everywhere) is also continuous everywhere na??
f(x) = 0 is a constant function...why shouldn't it be continuous?
Both the LHL and RHLs at x→0 are also 0...thus there is no breakage in the graph.
For a graph, it will be like f(x) = 2x with the part of x < 0 erased(as f(x) = 0 implies it is the X-axis).
39Arey...lol.
f(x) = 0 means X-axis....so for x < 0, the X-axis is your graph.
So for x < 0, the X-axis continues up till x = 0, where it tilts and becomes f(x) = 2x. There is no discontinuity.
If you have a doubt with the discontinuity part, put your finger on the X-axis and from -∞ take it till x = 0, from there go upwards on f(x) = 2x till +∞. You will see that you never had to take your finger even once off the curve(implying you were continuously tracing it). Hence there is no discontinuity anywhere.
39Lol...that was how our FIITJEE maths sir told us...a rudimentary way to check continuity and feel it for yourself if you have the graph :P