If $f$ is continuous on $[a,b]$ and $F(x) = \sup f([a,x])$. Prove that $F$ is continuous on $[a,b]$ .
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Supremum of a Continuous Function is Continuous
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If $f: \Bbb{R}\rightarrow\Bbb{R}$ is continuous at $0$ and $f(x)=f(2x)$ for each $x\in\Bbb{R}$ then $f$ is constant.
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How to prove that if $f$ is continuous a.e., then it is measurable.
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let $f \colon [a,b] \to \mathbb{R}$ be continuous on $[a,b]$ and..
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Proof of continuity $f(x+y) = f(x) + f(y)$
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Must $f$ be continuous?
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Proof that continuous function on $ [0, 2]$ has $f(x) = f(x-1)$ for some $ x$
151427
f, g continuous for all rationals follow by continuous on all reals?
145368
How to show $f:Y\to X^{+}$ is continuous?
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Show f is continuous on (0, ∞) iff f is continuous at 1 for a defined function.
117502
Proving that a certain continuous function is surjective.
77106
If $f$ is continuous and $g$ is integrable on $[a,b]$, with $g(x) \ge 0$ for all $x \in [a,b]$ ...
68048
Prove that if $f$ is differentiable on $[a,b]$ and $f'$ is monotonic on $[a,b]$ , $f'$ is continuous on $[a,b]$
14578
Proving that an additive function $f$ is continuous if it is continuous at a single point
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Let $S$ be a finite set. Show that any function $f: S \rightarrow \mathbb{R}^n$ is continuous on S. What if $S$ is countably infinite?
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Show that function $f$ has a continuous extension to $[a,b]$ iff $f$ is uniformly continuous on $(a,b)$
118207
Prove that the product of two continuous functions is continuous