= Method of continuity
{wiki}
The <method of continuity> transfers invertibility along a continuous family of <bounded linear operators> between fixed <Banach spaces>. Let $L_t:X\to Y$ be continuous in <operator norm> for $0\le t\le1$, let $L_0$ be invertible, and assume the uniform estimate
$$
\|u\|_X\le C\|L_tu\|_Y\qquad(u\in X,\ 0\le t\le1).
$$
Then every $L_t$ is invertible. The estimate gives injectivity and bounds any inverse by the same $C$.
For openness, if $L_s$ is invertible and $\|L_s^{-1}(L_t-L_s)\|<1$, a <Neumann series> inverts $I+L_s^{-1}(L_t-L_s)$. For closedness, suppose invertible $L_{t_j}$ approach $L_t$. For fixed $f\in Y$, put $u_j=L_{t_j}^{-1}f$. The uniform bound gives $\|u_j\|_X\le C\|f\|_Y$, and applying the estimate to $u_j-u_k$ gives
$$
\|u_j-u_k\|_X\le C\|(L_{t_k}-L_{t_j})u_k\|_Y\le C^2\|L_{t_k}-L_{t_j}\|\|f\|_Y.
$$
Thus $u_j$ converges in $X$ and its limit solves $L_tu=f$. The nonempty invertible parameter set is both open and closed in the connected interval.
For an <elliptic boundary value problem> with <Dirichlet boundary conditions>, one often uses $L_t=(1-t)\Delta+tL$ on the fixed space $C^{2,\alpha}$ with zero boundary values and target $C^{0,\alpha}$. A uniform <global Schauder estimate> together with a uniqueness/supremum estimate supplies the required bound. An estimate that still contains an uncontrolled solution norm on its right side is insufficient.<Lecture notes on Elliptic Partial Differential Equations by André Guerra and Gerard Orriols>[§4.5, Proposition 4.20 and equations (4.25)–(4.27)]{ref}
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