Is the following function continuous at #x=3# ?

Question

#f(x) = { (2, " if " x=3), (x-1, " if " x > 3), ((x+3)/3, " if " x < 3) :}#

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Answer 1 · 2017-03-04T08:35:51

Yes

Given:

#f(x) = { (2, " if " x=3), (x-1, " if " x > 3), ((x+3)/3, " if " x < 3) :}#

We find:

#lim_(x->3-) f(x) = lim_(x->3-) (x+3)/3 = (color(blue)(3)+3)/3 = 2#

#lim_(x->3+) f(x) = lim_(x->3+) (x-1) = color(blue)(3) - 1 = 2#

#f(3) = 2#

So the left and right limits agree and are equal to #f(3)#.

So this #f(x)# is continuous at #x=3#

graph{((x-3)/abs(x-3)+1)/2(x-1)+(1-(x-3)/abs(x-3))/2((x+3)/3) [-2.955, 7.045, -0.5, 4.5]}