Prove function is continuous at x = 0.5 and discontinuous at every other point
62947
If $f(x)$ is equal to $x$ or $1-x$ when $x$ is rational/irrational, then $f$ is continuous only at $x=1/2$
20273
Prove that if $|f(x)| \leq x^2$, then the function is continuous and differentiable at $x=0$.
12408
Prove that a function whose derivative is bounded is uniformly continuous.
4062
prove that $f(x)=\log(1+x^2)$ is Uniform continuous with $\epsilon ,\delta$ ...
153730
Show that $f(x)=x^2$ is continuous at $a=2$ using the $\delta-\epsilon$ definition of continuity.
88240
Two functions discontinuous, but sum continuous
79107
Give an example of a function $f: \mathbb{R} \to \mathbb{R}$ which is continuous only at $0$.
97553
Continuity of $\frac{2xy}{x^2+y^2}$ at $(0,0)$
44607
Prove that a continuous function on a closed interval attains a maximum
7358
For each $y \in \mathbb{R}$ either no $x$ with $f(x) = y$ or two such values of $x$. Show that $f$ is discontinuous.
154572
Continuity of a function at an isolated point
129856
Show that $f(x)=\frac{x}{1+|x|}$ is uniformly continuous.
125682
Proof verification: the product of two continuous functions is continuous
34940
Continuity of $f(x) = x^n$
126212
Proving uniform continuity on an interval
140445
Sequence of continuous functions whose pointwise limit is discontinuous
126842
Fixed point in a continuous function