Question

Prove cosxⁿ is continuous on R. (n ∈ N)

Answer 1

any function, f(x) is continuous at x = a only when $\displaystyle\lim_{h\to a^+}f(x)=\displaystyle\lim_{h\to a^-}f(x)=f(a)$

here, f(x) = cosxⁿ,
in case of cosine function, f(a+) = f(a-) = f(a)
for understanding let's take x = 0,
at x = 0, f(0^+) = $\displaystyle\lim_{h\to 0^+}cosx^n$=cos(0^n)= cos(0) = 1
f(0^-)=$\displaystyle\lim_{h\to 0^-}cosx^n$ = cos(0^n)=cos(0)=1
and f(0) = cos(0ⁿ) = cos0 = 1
hence , it follows the condition of continuity.
therefore , f(x) = cosxⁿ is continuous at x = 0
similarly, you can choose any point from all real numbers. you will get , f(x) follows the condition.
hence, finally we can say that f(x) = cosxⁿ is continuous on R.

method 2 :- any function is continuous on R when it must be function is differentiable on R.

here f(x) = cosxⁿ

differentiate with respect to x,

df(x)/dx = -sinxⁿ × d(xⁿ)/dx

= -sinxⁿ × nx^(n-1) , therefore f(x) is differentiable on R.
so, we can say f(x) is continuous on R.