How do you show that the function #f(x)=1-sqrt(1-x^2)# is continuous on the interval [-1,1]?

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Answer 1 · 2015-05-01T20:31:56

#f(x) = 1 - sqrt(1-x^2)# has domain #[-1,1]#

For #a# in #(-1,1)#, #lim_(xrarra) f(x) = f(a)# because

#lim_(xrarra) f(x) =lim_(xrarra)(1 - sqrt(1-x^2)) = 1-lim_(xrarra)sqrt(1-x^2)#

#= 1-sqrt(lim_(xrarra) (1-x^2)) = 1-sqrt(1-lim_(xrarra) x^2))#

#=1-sqrt(1-a^2) = f(a)#

So #f# is continuous on #(-1,1)#.

Similar reasoning will show that

#lim_(xrarr-1^+) f(x) = f(-1)# and

#lim_(xrarr1^-) f(x) = f(1)#

So #f# is continuous on #[-1,1]#.