I know that it may be a repeated question but that question failed to give me a satisfactory answer.Can someone provide with me an easy expiation for the question below
I know limit of function exists when Left hand limit = right hand limit
and a function is continuous only if Left hand limit = right hand limit= f(a)
Now,is there any case where Left hand limit = right hand limit ≠ f(a)
What's the physical meaning of limit of function.How is limit's definition different from continuous function's definition which states a function is continuos if any small change values of x will produce small change in value of y.
Let
$$f(x):=\begin{cases}x\ne0\to0,\\x=0\to1.\end{cases}$$
We do have $$\lim_{x\to0^+}f(x)=\lim_{x\to0^-}f(x)=\lim_{x\to0}f(x)=0$$ and the function is discontinuous at $0$.
In simple terms, a continuous function is one that doesn't have "jumps".