Proving a function is continuous for all real numbers This a homework question:
Prove  $f(x) = 2x^3 - 4x^2 + 5$ is continuous for all real numbers.
Which proof technique do I use for this? I don't really know where to start.

If you know how to prove that the identity function $f(x) = x$ is continuous, then by the algebra of continuous functions you have every polynomial continuous as they are just linear combinations of power functions i.e. $x^n$. If we have $f(x)=x$ continuous, then by the algebra of continuous functions $f\cdot f$ is continuous yielding $x^n$ continuous for any $n\in \mathbb{N}$ 
To show $f(x) = x$ is continuous consider 
$$|f(x_0)- f(x)|=|x_0-x|< \epsilon.$$
Hence, choosing $\delta := \epsilon$ gives the continuity of any polynomial function.
