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For what intervals is #f(x) = tan((pix)/4)# continuous?

Answer 1

Charles Arocha

#tan((pix)/4)# is continuous on intervals of the form
#(2 + 4k, 2 + 4(k+1)), k in ZZ#

#tan(x)# is continuous on #(pi/2 + kpi, pi/2 + (k+1)pi), k in ZZ#

So #tan((pix)/4)# is continuous where #(pix)/4# lies within such an interval. That is, where

#pi/2 + kpi < (pix)/4 < pi/2 + (k+1)pi#

Multiplying through by #4/pi# gives us

#2 + 4k < x < 2 + 4(k+1)#

Thus #tan((pix)/4)# is continuous on #(2 + 4k, 2 + 4(k+1)), k in ZZ#

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Answer 2

Charles Arocha

The function f(x) = tan((πx)/4) is continuous for all intervals where the tangent function is defined and does not have any vertical asymptotes or points of discontinuity. The tangent function is defined for all real numbers except for odd multiples of π/2. Therefore, f(x) = tan((πx)/4) is continuous for all intervals that do not contain odd multiples of π/2.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.