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Question
Show that the function f(x) = 2x - |x| is continuous at x = 0
Solution
`"L" ["f"(x)]_(x=0) = lim_(x->0^-) "f"(x) = lim_(h->0) "f"(0-"h")`
`= lim_(h->0) "f"(-"h")`
`= lim_(h->0) 2 (-"h") - |"h"|`
`= lim_(h->0) - 2"h" - "h"`
`= lim_(h->0) -3"h"`
= - 3(0) = 0 ....(1)
`"R" ["f"(x)]_(x=0) = lim_(x->0^+) "f"(x) = lim_(h->0^+) "f"(0 + "h")`
= `lim_(h->0) "f"("h") = lim_(h->0) 2"h" - |"h"|`
= `lim_(h->0) 2"h" - "h"`
= `lim_(h->0) "h" = 0` ...(2)
Also f(0) = 2(0) - |0| = 0
From (1), (2) and (3),
`"L" ["f"(x)]_(x=0) = "R" ["f"(x)]_(x=0)` = f(0) = 0
∴ f(x) = 2x - |x| is continuous at x = 0.
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