List of top Mathematics Questions on Continuity asked in UP Board XII

Prove that the function \( f(x) = |x| \) is continuous at \( x = 0 \) but not differentiable. 

Show that the function

\[ f(x) = \begin{cases} x^2 + 2, & \text{if } x \neq 0 \\ 1, & \text{if } x = 0 \end{cases} \]

is not continuous at \( x = 0 \).

Is the function \( f(x) \) defined by

\[ f(x) = \begin{cases} x + 5, & \text{if } x \leq 1 \\ x - 5, & \text{if } x > 1 \end{cases} \]

continuous at \( x = 1 \)?

Show that \( f(x) = |x| \, \textbf{is continuous at} \, x = 0.\)

Prove that \( f(x) = \tan x \) for all \( x \in \mathbb{R} \) is a continuous function.
Test whether the function $f : \mathbb{R} \to \mathbb{R}$ defined by \[f(x) = \begin{cases} x^3 + 3, & x \neq 0 \\ 1, & x = 0 \end{cases}\] is continuous at $x=0$.

If the function $f(x) = \begin{cases} ax+1, & x \leq 3 \\ bx+3, & x > 3 \end{cases}$ is continuous at $x = 3$, then find the values of $a$ and $b$.