Prove that cosx is continuous

For any real number c, we have limx→ccosx = cosc and cosx is defined for all real numbers. Thus, the function cosx is continuous everywhere.

Proof of cosx is continuous

Now, we will prove that cosx is continuous for all values of x by the epsilon-delta method. We will use the following two formulas:

  1. $\cos x -\cos y=2\sin \dfrac{x+y}{2} \sin \dfrac{y-x}{2}$
  2. $|\sin x| \leq |x|$

Prove cosx is continuous

Let f(x)=cosx and let x=c be an arbitrary real number.

Let ε>0 be any given positive number. We need to find a positive δ such that

|f(x)-f(c)| < ε whenever 0<|x-c|<δ

Choose δ=ε.

Now, for $x \in \{0<|x-c| <\delta=\epsilon\}$, we have that

$|f(x)-f(c)| = |\cos x -\cos c|$

$=2 |\sin \dfrac{x+c}{2} \sin \dfrac{c-x}{2}|$ by the above Formula 1.

$\leq 2 |\sin \dfrac{c-x}{2}|$ as we know that sinθ≤1 for all values of θ.

$\leq 2 |\dfrac{x-c}{2}|$ by the above Formula 2.

= |x-c|

< ε.

This shows that for any given ε>0, there exists a δ>0 such that whenever 0<|x-c|<δ, we have

|f(x)-f(x)| < ε.

Thus, by the epsilon-delta definition of continuity, we conclude that cosx is a continuous function at x=c. As c was an arbitrary real number, cosx is everywhere continuous.


Epsilon-Delta definition of Limit

Prove that sinx is continuous

Prove that modulus of x is continuous

Mod x-a is continuous at x=a but NOT differentiable


Q: cosx is continuous or not?

Answer: Yes, cosx is a continuous function, because limx→ccosx = cosc and cosx is defined for any real number c.

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