Determine if Continuous f(x)=tan((pix)/2)

Problem

ƒ(x)=tan((π*x)/2)

Solution

1. Identify the domain of the tangent function. The function tan(θ) is undefined whenever its argument θ is an odd multiple of π/2

2. Set up the equation for the points of discontinuity by setting the argument (π*x)/2 equal to the general form of these odd multiples.

(π*x)/2=π/2+n*π

3. Solve for x by dividing both sides of the equation by π/2

x=1+2*n

4. Conclude that the function is continuous on its domain, which consists of all real numbers except for odd integers. Since the function is not defined for all real numbers, it is not continuous everywhere.

Final Answer

ƒ(x)=tan((π*x)/2)* is continuous for all *x∈ℝ* except *x=2*n+1,n∈ℤ

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