Let $f: [0,1] \rightarrow \mathbb{R} $ be continuous with $f(0) = f(1)$ *note, there is a part b*
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Universal Chord Theorem
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continuous function: preimage of closure of a set is closure of preimage
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Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function. If $f(0)=f(1)$ then exists $x\in [0,1]$ s.t. $f(x) = f(x+\frac{1}{2})$
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Show a function for which $f(x + y) = f(x) + f(y) $ is continuous at zero if and only if it is continuous on $\mathbb R$
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Continuous mapping $f: [0,1]\rightarrow (0,1)$ CSIR December $2013$
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Proof if $f$ continuous in $x_0$ then there is a neighbourhood of $x_0$ so f bounded.
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Continuity of a Function in Terms of Closure and Interior
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Let $f$ be continuous on $ [a,b]$, differentiable on $ (a,b) $ and $ f(x) \neq 0 \forall x \in (a,b) $ Prove $ \exists c \in (a,b) $ such that
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Continuity of $f(x) = x^n$
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Show that f(x) defined on [0,infinity) R to is continuous at 0.
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$f:X\longrightarrow\mathbb{R}$ is continuous iff $\{x\in X:f(x)\geq\alpha\}$ and $\{x\in X:f(x)\leq\alpha\}$ are closed $\forall\alpha\in\mathbb{R}$
132013
How do I show that if $f : [0,1] \to [0,1]$ is continuous, then there is some $c$ such that $f(c) = c$?
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If $\{x:f(x) > a\}$ and $\{x:f(x) < a\}$ are open$\forall~a\in\mathbb{R}$ then f is continuous.
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Remark from baby Rudin about continuous function
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Linear functional $f$ is continuous at $x_0=0$ if and only if $f$ is continuous $\forall x\in X$?
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Extension of continuous function on a closed subset of $\mathbb{R}$
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Questions about $f: \mathbb{R} \rightarrow \mathbb{R}$ with bounded derivative