Analytic off the real axis

Question

If $f:\mathbb C \longrightarrow \mathbb C$ is continuous and $f$ is analytic off the real axis, then show that $f$ is entire.

Answer 1

By continuity, $f$ is bounded on $\{x+iy: -a\leq x\leq a, -\varepsilon \leq y \leq \varepsilon\}$. Then by dominated convergence, $\lim_{y\to 0}\int_{-a}^af(x+iy)\ dx=\int_{-a}^af(x)\ dx$.

Now we can take the integral of $f$ on a cirlce of radius $a$ centered at $0$ as the limit of the integrals over two semi-circles, one in each half-plane, and apply Morera's theorem.

Answer 2

Theorem: (Symmetry principle) If $f^+$ and $f^−$ are holomorphic functions in $Ω^+$ and $Ω^−$ respectively, that extend continuously to I and $$f^+(x) = f^−(x)\quad for\;all\;x ∈ I,$$ then the function $f$ defined on $Ω$ by $$f(z) = $$$$f^+ (z)\quad if\quad z ∈ Ω^+ ,\quad f^+(z) = f^−(z)\quad if\quad z ∈ I,\quad f^−(z)\quad if\quad ∈ Ω^−$$ is holomorphic on all of $Ω$.(COMPLEX ANALYSIS Elias M. Stein & Rami Shakarchi page 58)

According to the theorem you have $f^+=f^-=f$ , $Ω^+$ is upper half plain $Ω^-$ is lower half plain and $I=\mathbb{R}$ so $f$ is holomorphic over $\mathbb{C}$ means entire.