Are The Continuous Functions On $[0,1]$ Countable?

Probability Level 1

Let $S$ denote the set of continuous functions $f: [0,1] \to \mathbb{R}$ .

Which of the following is true of $S?$

S

is finite

S

is countably infinite

S

is uncountably infinite

1 solution

Guillermo Templado
Mar 4, 2016

Every constant function $f:[0,1] \rightarrow \mathbb{R}$ is a continuous function. Since cardinal $\mathbb{R}$ is uncontable infinite then cardinal S is uncontable infinite ( $|S| \ge |\mathbb{R}|$ )

Are The Continuous Functions On [ 0 , 1 ] [0,1] [ 0 , 1 ] Countable?

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Are The Continuous Functions On $[0,1]$ Countable?