$f|_{A_i} $is continuous, $A_i$ closed $\implies$ $f:X\to Y$ is continous
51156
Function is Continuous When Restricted to Closed Sets Which Cover the Space
143174
property of continuous function
80643
Show that f(x) defined on [0,infinity) R to is continuous at 0.
74364
Show that function $f$ has a continuous extension to $[a,b]$ iff $f$ is uniformly continuous on $(a,b)$
31266
Proof of "the continuous image of a connected set is connected"
69716
$f_x$ is Borel measurable and $f^y$ is continuous then $f$ is Borel measurable
158423
How to prove that if $f$ is continuous a.e., then it is measurable.
155618
Slightly changing the formal definition of continuity of $f: \mathbb{R} \to \mathbb{R}$?
149354
Proof of continuity $f(x+y) = f(x) + f(y)$
14578
Proving that an additive function $f$ is continuous if it is continuous at a single point
118332
proof $f(\overline{A})\subseteq \overline{f(A)} \Leftrightarrow f$ continuous
127868
If $f$ is nonnegative and integrable then $F(x) = \int_{-\infty}^x f$ is continuous.
55368
$f$ is continuous, if $f_n$ continuous and $f_n\to f$ uniformly
32359
Inner product of two continuous maps is continuous
22061
$|f(x)-f(y)|\geq k|x-y|$.Then $f$ is bijective and its inverse is continuous.
53627
Proving continuity of $f$
100740
proof of that a continuous function has a fixed point