Assume h is an increasing function from the set of real numbers to the set of reals. Show h(R) connected implies that h is continuous.
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Increasing, continuous function implies connectivity and viceversa.
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Continuity in Slices + Mapping Compact Sets to Compact Sets = Continuous
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Increasing real valued function whose image set is connected
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Must $f$ be continuous?
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if $f$ is continuous on $\mathbb{R}$ and $f(r)=0,r \in \mathbb{Q}$, then $f(x)=0,x \in \mathbb{R}$
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$f(\mathbb{R}\setminus \mathbb{Q}) \subseteq \mathbb{Q}$ and $f(\mathbb{Q}) \subseteq \mathbb{R}\setminus \mathbb{Q}$ imply that $f$ is not continuous
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Proving that an additive function $f$ is continuous if it is continuous at a single point
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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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Continuity of a function of product spaces
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Proof that continuous function on $ [0, 2]$ has $f(x) = f(x-1)$ for some $ x$
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Continuity of a function to the integers
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Real Analysis Question: continuity of $f(x) = \sum_{q_n<x} 1/n^2$
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If $f\colon \mathbb{R} \to \mathbb{R}$ is such that $f (x + y) = f (x) f (y)$ and continuous at $0$, then continuous everywhere
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How to show $f$ is continuous at $x$ IFF for any sequence ${x_n}$ in $X$ converging to $x$ the sequence $f(x_n)$ converges in $Y$ to $f(x)$
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proof $f(\overline{A})\subseteq \overline{f(A)} \Leftrightarrow f$ continuous
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Show that if $f:X\to Y$ is a continuous function if and only if the graph of $f$ is a closed subset of $X\times Y$
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For closed subsets $A,B \subseteq X$ with $X = A \cup B$ show that $f \colon X \to Y$ is continuous iff $f|_A$ and $f|_B$ are continuous.