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How to show a function is continuous in a given interval?

This might be a very basic question but I'm very new to calculus and I intend to prove that a given function is continuous in a given closed interval. How can I do it (what arguments are needed to say that the function is continuous in that interval)? I know about limits and derivatives.

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  • Using limits, a function $f:[a,b]\longrightarrow \mathbb R$ is continuous if, for every $c \in [a,b]$, $\lim_{x\longrightarrow c} f(x) = f(c)$. – Luis Dias Mar 05 '23 at 14:27
  • @LuisDias thank you. Is there a way to approach continuity through derivatives? – Diff_Equations Mar 05 '23 at 14:29
  • Alternatively, $f:[a,b] \longrightarrow \mathbb R $ is continuous if, for every $x \in [a,b] $, $f'(x)$ is defined. – Luca Armstrong Mar 05 '23 at 14:32
  • @LucaArmstrong Your second comment is somewhat misleading, because a function can be continuous at a point, without being differentiable at that point. For example, the function $y = |x|$ is continuous at $x=0$ but the derivative at $x = 0$ does not exist. – user2661923 Mar 05 '23 at 15:43
  • Let $A$ represent the domain of the function $f, ~[a,b]$. The continuity definition that I was taught is that $f(x)$ is continuous at $~x_0 \in A~$ if and only if, for all $\epsilon > 0$, there exists a $\delta > 0$ such that for each $~x \in A~$ such that $~0 < |x-x_0| < \delta,~$ you have that $|f(x) - f(x_0)| < \epsilon.$ – user2661923 Mar 05 '23 at 15:49
  • @user2661923 Yes, sorry for the confusion. I meant that if $f$ has a defined derivative at some point $x\in [a, b]$, this implies continuity at that point. – Luca Armstrong Mar 05 '23 at 16:24

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