Exercise 4.1

Question

Exercise 1:
    Suppose $f$ is a real function defined on $\mathbf R$ which satisfies $$\lim_{h\rightarrow 0}\bigl[f(x+h)-f(x-h)\bigr]=0$$ for every $x\in\mathbf R$. Does this imply that
    $f$ is continuous?

ghostofgarborg · Accepted Answer

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No. As an example, take the function 
    $$ f(x) = \begin{cases} 1 & x = 0 \ 0 & \text{otherwise} \end{cases} $$
    which is discontinuous at $0$, although $\lim_{h \to 0} [f(x+h) - f(x-h)] = 0$ everywhere.

Amit Awasthi · Answer

\begin{claim}
      $f$ is not necessarily continuous.
    \end{claim}
    \begin{proof}
      We can separate the equation into two limits to give:
      \begin{equation*}
        \lim_{h \rightarrow 0} f(x+h) = \lim_{h \rightarrow 0} f(x-h)
      \end{equation*}
      However, this does not imply continuity since $f(x)$ need not equal the limits on either side.
    \end{proof}

Exercise 4.1

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Exercise 4.1

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