Question

Prove cosⁿx 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) = cosⁿx,
in case of cosine function, f(a+) = f(a-) = f(a)
for understanding let's take x = 0,
at x = 0, f(0^+) = $\displaystyle\lim_{h\to 0^+}\cos^nx$=(1)ⁿ = 1
f(0^-)=$\displaystyle\lim_{h\to 0^-}\cos^nx$ = (1)ⁿ = 1
and f(0) = cosⁿ0 = (1)ⁿ = 1
hence , it follows the condition of continuity.
therefore , f(x) = cosⁿx 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) = cosⁿx is continuous on R.