Some continuity questions

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Question from Mac, a student:

Can you please help me out to solve this.

problem 1
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which of the following statement is true or false ?
a) f(x) = x + [x], x is the member of Z is not continuous at x=0
b) lim x->0+ (f'(x)) = lim x->0- (f'(x))

my doubt here is, what is that [x] in that function ?

problem 2
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f(x) = [x] + 1 over positive integer including 0. what is the total number of
point of discontinuities of f(x) ?

Again this [x] confuses me, because if i take [x] this as |x|, then this function is
continuous. can you please help me out ?

Hi Mac,

The notation [x] confuses me also. One thing I can tell you is that if Z is the integers then the function in part a) can not be continuous at 0. If the function is to be continuous then the limit as x approaches zero must exist. This requires that the function be defined at all points in some small interval around zero. But this can't be true since the function is only defined at integer points.

For problem 2 I agree that if [x] = |x| then the function is continuous at all positive values of x and continuous from the left at x = 0.

Harley Weston

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.