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

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Answer 1 · 2015-11-13T05:33:19

$tan((pix)/4)$ 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
$(2 + 4k, 2 + 4(k+1)), k in ZZ$

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