A function f(x) is defined by the following formulas
f(x) = $x^2$ + 1 when x is irrational,
tan(x) when x is rational.
At how many x in the interval [0, 4$\pi$] is f(x) continuous? on solving i hope there is 4 x in the interval.but while i try to get use definition of continuity I'm not sure whether i done correctly.Any answer using $\varepsilon$ and $\delta$ method or sequential definition of continuity is appreciable
You are right in saying that $f$ will be continuous at $x$ if and only if $\tan(x)=x^2+1$. Sequential continuity is the same as continuity for such functions.
The next task is to count the number of solutions of that equation. Let $g(x)=x^2+1$. First on the interval $[0,\pi/2[$ we have $\tan(0)=0$ and $g(0)=1$, but $\tan(x)\rightarrow\infty$ as $x\rightarrow\left(\dfrac{\pi}{2}\right)^-$ while $g$ stays bounded. You can summon the intermediate value theorem to show that there is a solution to $\tan(x)=x^2+1$ on the interval $[0,\pi/2[$.
Apply a similar reasoning to the intervals $]\pi/2,3\pi/2[$, $]3\pi/2,5\pi/2[$ and $]5\pi/2,7\pi/2[$. Clearly there is no solution on $]7\pi/2,4\pi]$ (Why?).