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

1 Answer
sente
Nov 13, 2015

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

Explanation:

#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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